In your Georgia State interview, you should have been asked for 'taxes paid to another state'.
Be sure that you entered the amount of tax paid to New York here, so that Georgia gives you tax credit for that amount.
If the New York tax was on a W-2, TurboTax may have picked it up, but it is always best to make sure.
You are not being taxed more because you made income in New York. Maybe you didn't have enough tax withheld on your W-2's?
Since we can't see your return in this forum, I suggest you review your Georgia return and worksheets to see how the tax was calculated for you.
You can get your returns reviewed by TurboTax Live if you want individualized help.
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