New York has a rule that the state wages in box 16 of your W-2 must be the same as the federal wages in box 1. That doesn't mean that you actually earned all the income in New York. On your New York tax return you will show how much was actually New York income.
Do not enter the W-2s as two separate W-2s in TurboTax. Enter the New Jersey W-2, and add the NY line of state information, boxes 15, 16, and 17 as a second line of state information on the same W-2.
When you do your New York tax return, you will get a screen with the heading "Your Form W-2 Summary." Click the Edit button for the W-2. It will ask you to select an allocation method. Select "Allocate by Percentage." On the next screen enter zero as the New York State Percentage. This will give you a full refund of all the New York tax that was withheld.
On your New Jersey tax return you will not get the credit for income tax paid to other jurisdictions because you are not paying any New York tax.
