The income you earned for working in New York is New York income, not New Jersey income. New Jersey taxes your New York income because you are a New Jersey resident (but you get a credit on your New Jersey tax return for part of the tax that you pay to New York). I think technically your W-2 should have a New Jersey line with only the additional amount that is taxed by New Jersey but not by New York. But employers practically never do this. They typically either do what your employer did, and report only New York income, or they report the full amount of your income on both the New York and New Jersey lines. The trouble with that second approach is that if you enter the W-2 that way in your tax software (any tax software) your New Jersey tax return ends up calculating tax on both the New York and New Jersey amounts. In other words, you get taxed by New Jersey on double your actual income.
It's a messy situation, and neither state has ever given employers clear instructions for how to fill out the W-2. I suggest that you just enter your W-2 in TurboTax the way it is, and don't worry about it.
