I move to NJ in Sep 2023 but still work in pa. My employee withheld both NJ tax and PA tax since Sep 2023 so that My W2 shows two state income tax

1. Yes, if during 2023 you lived in NJ and PA, you need to file part-year resident return both states. New Jersey does not have a special form for part-year filers, you must use the regular resident return, Form NJ-1040. The return provides a line for you to show the period of your residency in the state.

For NJ income filing requirements, check: NJ Treasury

2. Income earned while living in NJ and working in PA is taxable on you NJ return because you were a resident of NJ and taxable by PA since you were working in PA. 

However, as NJ Part-year resident, you may be able to claim a credit for taxes paid to PA. To avoid double taxation on income earned while living in NJ and working in PA. 

3. Income earned before September while working and living in PA is only taxable on your PA return. 

In TurboTax, you need to make sure you've set up your personal information correctly and, you'll need to allocate your income between the two states. 

Please check: How do I file a part-year state return?

 How do I allocate (split) income for a part-year state return?

[Edited 03/25/2024, 09:52 AM,PST]