Is it normal in USA/UK to use the phone while driving?

In the UK, it is illegal to use your phone while driving unless it's an emergency and you're calling the emergency services (999), from the UK government website:

It’s illegal to ride a motorcycle or drive using hand-held phones or similar devices.

You can get an automatic fixed penalty notice if you’re caught using a hand-held phone while driving or riding. You’ll get 3 penalty points on your licence and a fine of £100.

Your case could also go to court and you could be disqualified from driving or riding and get a maximum fine of £1,000.

If you’re the driver, you can only use your phone in a vehicle if you need to call 999 in an emergency and it’s unsafe or impractical to stop

In the US, it's generally legal though it's up to the state government, however in California, which the first 6 seasons were primarily set in, it's illegal.

However, the scene before the advert break in which you last see Jack before the driving scene you mention, he's at the CIA station in London with Kate, so I think it's a fair assumption that he's in a CIA (or US government/embassy) issued vehicle, which the police would be able to tell from the automatic license plate readers, so it's quite unlikely he'd actually be pulled over for it, unless of course it will be used to add dramatic tension to a chase scene in the next episode.