Have you spoken to your PB for an update? 4 weeks is not unheard of, but every municipality is different. I would go on-line and check with your local building department. They may have a guideline as to the approximate timeframe for permit approval. You might also be able to check on-line to see if the permit request has been submitted. Good Luck!
The answer is...It depends. It depends on your county, as well as your PB. I signed my contract in November 2016. My PB had to wait on the screen enclosure company to provide them with the engineering plans. The screen company of course was not very "speedy", and then throw the holidays into the mix...needless to say it was not submitted in 2016. On the Treasure Coast, the average time for pool permitting can take anywhere from 4-6 weeks. My permit was submitted on 1/9/17 and issued on 1/30/17. The PB started the dig on 2/3/17. Check out the website of your local building department, there is usually a lot of helpful information on there in regards to permitting. I hope this helps!