Is this sentence correct?

It isn't grammatically correct due to the past perfect.

You should know that the word "after" is almost always followed by a past perfect while the word before is always followed by a simple past.

Your sentence should be written in this way: Jack and Emily flew back to New York immediately after they had gone to the bar.

It means that the action to go to the bar happened before flaying back to New York.

However if you want to replace the word after with before the sentence above should be written in the following way: Before Jack and Emily flew back to New York they had gone to the bar. (The word immediately could be deleted since the phrase means the same even though you don't use it)

Hope it helps.

No it's not correct.

The correct way to say this is -

Jack and Emily had flown back to New York immediately after HAVING BEEN to the bar.

As soon as Jack and Emily returned New York , they immediately headed for the bar OR Immediately went to the bar.

Hope this helps

Jack and Emily immediatetly flew back to N.Y. after they had gone to the bar. ..OR:

After they had gone to the bar, Jack and Emily immediately flew back to N.Y.

Two events taking place one after another in the past, therefore the first one should be in past perfect, while the second one should be in simple past.

Another confusing thing is that in your original sentence, you stated the second event first. Therefore, from my above choices the second sentence is preferred.

Seems a little awkward. I think it's because of the word immediately.