TL;DR: Discriminatory policies against Jews came long before the death camps, and the US was well aware of these policies by the mid-30's. The US government became aware of internment of Jews soon after that policy was enacted as well. The existence of death camps came to the attention of the Allied governments by 1942 at the latest, but wasn't fully believed at first. By the time of the Normandy landings, though, many average citizens of the US - especially American Jews - were well aware of the rumors about such facilities.

Before the holocaust:

The US was aware of the Nazis' oppression and brutality towards the Jews soon after Hitler took power.

In 1933, new German laws forced Jews out of their civil service jobs, university and law court positions, and other areas of public life. In April 1933, laws proclaimed at Nuremberg made Jews second-class citizens. These Nuremberg Laws defined Jews, not by their religion or by how they wanted to identify themselves, but by the religious affiliation of their grandparents. Between 1937 and 1939, new anti-Jewish regulations segregated Jews further and made daily life very difficult for them. Jews could not attend public schools; go to theaters, cinema, or vacation resorts; or reside or even walk in certain sections of German cities.
- Source

The US, and the other future Allied powers, did little to oppose the early signs of anti-Jewish policies in Nazi Germany.

In 1933 the Nazi Party under the leadership of Adolf Hitler legally gained power in Germany, and it did not take long for the persecution of Jews to begin. On April 7, 1933, the Reichstag, under Hitler’s influence, adopted the Restoration of the Professional Civil Service Law that dismissed non-Aryans from governmental positions.

This restoration of this act was more significant than it appeared because the Civil Service decree included bankers, lawyers, railroad and hospital workers, medicine, law, schools, universities, and the arts. The Nazi intent of these laws was to eliminate Jewish influence in Germany.

Not long after the Nazis implemented the Civil Service laws in Germany the first significant protest against Jewish mistreatment came from the American Jewish Committee. This was a committee developed to protect the rights of Jews, and they asked the American government to properly investigate what was happening in Germany. The American Jewish Congress began protests and had a strong membership and following that would help inform the rest of the country of the worsening conditions in Germany. For the United States to protest another country’s treatment of its citizens during the era of segregationist Jim Crow laws was somewhat hypocritical. The United States had no basis to criticize another country for racial injustice when it practiced similar discrimination of certain races of people. Nonetheless protests began and Christian groups, such as the American Christians, made public protests against Nazi anti-Semitic acts with the support of influential public figures such as Alfred E. Smith, Newton D. Baker, and John W. Davis. Slowly information about Nazi anti-Semitic acts began to filter into the United States. The State Department after hearing of these allegations asked the American Embassy in Berlin to investigate these accusations. The embassy liaison reported that there was good indication that the Nazis were hiding something. When the Germans were questioned on this issue of Jewish mistreatment, the Nazis claimed this anti-Semitic rhetoric and attacks were isolated incidents and they were in process of being stopped. This was very early in Hitler’s regime and the true magnitude of his hatred of Jews was not yet identified. Ranking United States government individuals probably knew that Hitler inherently disliked Jews because of his biography Mein Kamp, but there was no precedent of Jewish genocide to use as a guide to future events.

Throughout 1933 reports of Jewish mistreatment at the hands of the Nazis kept surfacing. H.R. Knickerbocker, who was the New York Evening Post correspondent in Berlin, reported in April 1933 that an undetermined number of Jews had been killed, or fled, or been deprived of their livelihood in the Reich. In this Newsweek article Mr. Knickerbocker made two interesting observations. The first was that he did not understand why the Nazis had such hatred for the Jews, and secondly he concluded that the Germans were envious of Jewish accomplishments in Germany. These two conclusions would prove to be very astute in the years that followed, yet few people in the United States seemed to understand what was happening to the Jews in Germany. From the everyday citizen to the higher powers in the government this hatred was not fully understood until it was too late. The Nazis under Hitler’s leadership ignited the hidden anti-Jewish attitudes of gentiles in Germany. Hitler was a master orator and he convinced the German citizens that the Jews were the reason that Germany lost World War I and fell into economic despair in the 1920s.
- Source

The easiest way for other countries to help the Jews of Germany was to increase immigration quotas, but this was not politically feasible for many years, because the world - including the US - remained mired in the Great Depression, and public sentiment was focused on finding jobs for American citizens at the expense of all other concerns. Furthermore, Anti-Semitism was rampant across Europe and the US at the time, so sympathy for the Jews of Germany was fairly hard to find.

Many German and Austrian Jews tried to go to the United States but could not obtain the visas needed to enter. Even though news of the violent pogroms of November 1938 was widely reported, Americans remained reluctant to welcome Jewish refugees. In the midst of the Great Depression, many Americans believed that refugees would compete with them for jobs and overburden social programs set up to assist the needy.
- Source

And:

Like most other countries, the United States did not welcome Jewish refugees from Europe. In 1939, 83% of Americans were opposed to the admission of refugees.3 In the midst of the Great Depression, many feared the burden that immigrants could place on the nation’s economy; refugees, who in most cases were prevented from bringing any money or assets with them, were an even greater cause for concern. Indeed, as early as 1930, President Herbert Hoover reinterpreted immigration legislation barring those “likely to become a public charge” to include even those immigrants who were capable of working, reasoning that high unemployment would make it impossible for immigrants to find jobs.

While economic concerns certainly played a role in Americans’ attitudes toward immigration, so too did feelings of fear, mistrust, and even hatred of those who were different. Immigration policies were shaped by fears of communist infiltrators and Nazi spies. Antisemitism also played an important role in public opinion.
- Source

For these reasons, many historians rightly assert that other European nations, and the US, share some of the blame for the holocaust.

During the holocaust:

By 1940, reports on Auschwitz in particular were arriving in London, via the Polish resistance and the Polish Government-in-Exile.

In 1942, the Polish Government-in-Exile submitted a paper describing the mass extermination of Jews in occupied Poland. It was not the first such paper they had produced, but it was the first to be made public.

Note to the Governments of the United Nations - December 10th, 1942

Ponizszy dokument informuje wszystkie Narody Zjednoczone o exterminacji Zydow polskich i europejskich przez Hitlerowcow na terenach okupowanych.

INTRODUCTORY NOTE
The purpose of this publication is to make public the contents of the Note of December 10th, 1942, addressed by the Polish Government to the Governments of the United Nations concerning the mass extermination of Jews in the Polish territories occupied by Germany, and also other documents treating on the same subject.
- Source

A week later, the New York Times covered the paper:

ALLIES CONDEMN NAZI WAR ON JEWS; United Nations Issue Joint Declaration of Protest on 'Cold-Blooded Extermination' 11 ALLIES CONDEMN NAZI WAR ON JEWS
December 18, 1942

WASHINGTON, Dec. 17 -- A joint declaration by members of the United Nations was issued today condemning Germany's "bestial policy of cold-blooded extermination" of Jews and declaring that "such events can only strengthen the resolve of all freedom-loving peoples to overthrow the barbarous Hitlerite tyranny."
- NY Times

Here is the full text of the Joint Declaration by the UN:

"The attention of the Belgian, Czechoslovak, Greek, Jugoslav, Luxembourg, Netherlands, Norwegian, Polish, Soviet, United Kingdom and United States Governments and also of the French National Committee has been drawn to numerous reports from Europe that the German authorities, not content with denying to persons of Jewish race in all the territories over which their barbarous rule has been extended, the most elementary human rights, are now carrying into effect Hitler's oft-repeated intention to exterminate the Jewish people in Europe.

From all the occupied countries Jews are being transported in conditions of appalling horror and brutality to Eastern Europe. In Poland, which has been made the principal Nazi slaughterhouse, the ghettos established by the German invader are being systematically emptied of all Jews except a few highly skilled workers required for war industries. None of those taken away are ever heard of again. The able-bodied are slowly worked to death in labor camps. The infirm are left to die of exposure and starvation or are deliberately massacred in mass executions. The number of victims of these bloody cruelties is reckoned in many hundreds of thousands of entirely innocent men, women and children.

The above-mentioned governments and the French National Committee condemn in the strongest possible terms this bestial policy of cold-blooded extermination. They declare that such events can only strengthen the resolve of all freedom-loving peoples to overthrow the barbarous Hitlerite tyranny. They reaffirm their solemn resolution to insure that those responsible for these crimes shall not escape retribution, and to press on with the necessary practical measures to this end."
- Wikipedia

In May of 1943, an Allied conference in Bermuda decided not to increase immigration quotas in the US and UK for European Jews fleeing Nazi occupation, effectively condemning many people to death in the camps. The next week, the American Zionist Committee for a Jewish Army published a piece in the New York Times, denouncing the decision:

By this point, the Allies' initial reaction - dismissing the reports of mass extermination as Polish propaganda - was untenable. For the past year, the Allied intelligence services had been intercepting and decoding German police communiques detailing intake and death records from many concentration camps. These intercepted messages are known as the Bletchley Park Concentration Camp Decodes.

The cat being out of the bag, the mainstream press began reporting on Nazi policies of mass extermination more frequently after 1943.

Conclusion:

It would have been possible for an average citizen/soldier from the US to remain mostly unaware of the holocaust, but almost everyone would have been aware of Nazi persecution of Jews by 1944, and many would have heard at least some rumors or reports of mass extermination. American Jews would have been much more acutely aware of these stories, and would be likely to treat the Nazis as their enemies on a deeply personal level - much more so than gentile Americans, who were more opposed to Japan than Germany, did.

This seems more like a question that belongs on History SE, but I'll throw my $.02 in here...

TLDR: As early as 1933 but certainly by Nov. 24, 1942.

There is much debate on the topic of your question. The following is from the Holocaust Encylopedia

US STATE DEPARTMENT RESPONSE TO NEWS OF THE “FINAL SOLUTION”

In August 1942, the State Department received a report sent by Gerhart Riegner, the Geneva-based representative of the World Jewish Congress (WJC). The report revealed that the Germans were implementing a policy to physically annihilate the Jews of Europe. Department officials declined to pass on the report to its intended recipient, American Jewish leader Stephen Wise, who was President of the World Jewish Congress.

Despite the State Department's delay in publicizing the mass murder, that same month Wise received the report via British channels. He sought permission from the State Department to make its contents public. Undersecretary of State Sumner Welles asked Wise not to publicize the information until the State Department confirmed it. Wise agreed and after three months the State Department notified him that its sources had confirmation. On November 24, 1942, Wise held a press conference to announce that Nazi Germany was implementing a policy to annihilate the European Jews. A few weeks later, on December 17, the United States, Great Britain, and ten other Allied governments issued a declaration denouncing Nazi Germany's intention to murder the Jews of Europe. The declaration warned Nazi Germany that it would be held responsible for these crimes.

US PRESS COVERAGE OF THE “FINAL SOLUTION”

During the era of the Holocaust, the American press did not always publicize reports of Nazi atrocities in full or with prominent placement. For example, the New York Times, the nation's leading newspaper, generally deemphasized the murder of the Jews in its news coverage. The US press had reported on Nazi violence against Jews in Germany as early as 1933. It covered extensively the Nuremberg Laws of 1935 and the expanded German antisemitic legislation of 1938 and 1939. The nationwide state-sponsored violence of November 9-10, 1938, known as Kristallnacht (Night of Crystal), made front page news in dailies across the US as did Hitler's infamous prediction, expressed to the Reichstag (German parliament) on January 30, 1939, that a new world war would mean the annihilation of the Jewish “race.”

As the magnitude of anti-Jewish violence increased in 1939-1941, many American newspapers ran descriptions of German shooting operations, first in Poland and later after the invasion of the Soviet Union. The ethnic identity of the victims was not always made clear. Some reports described German mass murder operations with the word "extermination." As early as July 2, 1942, the New York Times reported on the operations of the killing center in Chelmno, based on sources from the Polish underground. The article, however, appeared on page six of the newspaper. Although the New York Times covered the December 1942 statement of the Allies condemning the mass murder of European Jews on its front page, it placed coverage of the more specific information released by Wise on page ten, significantly minimizing its importance.

It was a combination of getting through Hell (can you imagine running up that beach with people dying left and right?!?) and living, as well as the fact that he had just helped break through a major line of defense and defeated people who were responsible for hundreds of thousands (at that point) of his people's deaths.

I'm no psychologist (though I did stay at a Holiday Inn once), but I think once the adrenaline of getting through that beach wore off, all the fear and anger hits you and it's going to cause some emotions. If you're not emotional after that, you're numb.