Why aren't there any visitor logs?

I'm aware a lot of the comments discuss this already, but this really does appear to be a case of bad program design. Pick any movie discussing website online and you can see people echoing your views:

IMDB:

...if any government department in any country ran anything remotely as incompetently as these guys do in "The Following" then we are seriously, seriously doomed.

And then they escape because the FBI or the Police were looking the other way at the critical moment.

...Joe's escape from jail was the final straw. The premise to be moved to another jail is bizarre and if that's not bad enough then prison officials, FBI agents and Federal marshals all manage not to be able to put one prisoner in the back of a transport truck and move him. But he then gets magically in the trunk of his lawyer's car and escapes. We all have different tastes in shows but I find it baffling that anyone could find this show even remotely watchable because it's so insulting to the viewers.

From Digital Spy, who note the lack of a prison log of any sort:

...There's Ryan's slipper snake ex Molly (Jennifer Ferrin) - who's now one of Joe's followers. A self-styled 'Angel of Death', Molly has it in for Hardy and first met Carroll - you guessed it - in prison. Seriously, Feds, check that visitor log.

And finally, a rather entertaining rant over at Time Entertainment. Some of the choice quotes from this article:

The good guys are apparently idiots.

At first, I was okay with this. I could accept that law-enforcement might be fallible; otherwise, the show would be an hour of Kevin Bacon’s former FBI agent Ryan Hardy talking to James Purefoy’s creepy-yet-charming psychopath Joe Carroll, explaining that there was no way that an imprisoned serial killer would’ve been able to use the Internet to set up thousands of websites to gain acolytes without everyone in the prison knowing about it (and logging the IP addresses of those visiting the sites).

...The problem with The Following is that these moments of idiocy are not few and far between. In fact, as Vulture pointed out earlier this week, every single episode of the show has asked us to believe that the FBI agents hunting the serial killing cult have made at least one fatal, and often ridiculous, error in judgment or procedure that prevents them from catching the bad guys...

So effectively, the answer to your question appears to be because the FBI, by the decision of the show's creators, are complete idiots - something many people online have noticed and been irritated by.

If I recall correctly, they actually do have a visitor's log, but those visitors were provided with fake IDs from one of the cult members. This way, his visitors couldn't be rounded up once the Feds were on to the plot.

There's a fair amount of "suspension of belief" at play in this show. You are asked to believe a lot of stuff that just isn't rationally believable, and the police/Feds are definitely made out to be fallable. But, then again, almost any Action movie really uses that same concept. The good guys are always smarter than the bad guys. Chuck Norris, while indeed God-like, most likely wouldn't get past the 3rd armed guard in real life, nor would John Rambo. But you have to buy that stuff in order to keep the ride going. So, in The Following, the FBI has to be incompetent enough to keep the plot moving.

Personally, after the back-and-forth about the kid getting away and then the wife gets captured, then the wife gets away and the kid get captured again, it all got too crazy. The first couple episodes were some of the best TV I've ever seen, but then I got tired of suspending my belief that much.

The lack of visitor logs, necessary documentation to visit a prisoner and in an age of relatively inexpensive HD video, clear images of Joe Carroll's visitors, are part of program's necessary suspension of disbelief that allows its fictional narrative to be sustained. The series would not be able to function or would do so poorly if common sense security measures were employed. Or the level of writing would have to improved to compensate for these glaring errors.

Since the series is doing well for the network, filling any plot holes is a low priority. However, if the program begins to falter in the ratings or if the show's run extended further into the future than anticipated, attempts may be made to close the loopholes or fill in the gaps in the plot's logical inconsistencies.