You are correct that the city income tax refunds for 2021 and 2022 are not taxable if you took the standard deduction on your federal tax returns for 2021 and 2022. If you know that the refunds are not taxable, the simplest thing to do is to not enter them in TurboTax at all. There is no requirement to enter them.
TurboTax does not assume that the 2021 city tax refund is taxable. It does not calculate the taxable amount for years earlier than 2022. When you entered the refund, if you indicated that it was for 2021, TurboTax asked you to enter the taxable amount, which you have to calculate yourself. The screen where it asks is not clearly worded, but it tells you to enter zero if you did not itemize on your federal tax return for that year. That screen initially has the full amount of the refund entered as taxable. You are supposed to change it to zero if none of it is taxable. Because of the unclear wording, you might have misunderstood the screen and clicked Continue without changing the taxable amount to zero.
