Any income that you earned from working in New York is subject to New York tax, even if you lived in a different state.
Look at the actual New York state tax form, not the summary in TurboTax. As a part-year resident of New York, you should be filing Form IT-203. The first page of Form IT-203 has two columns for income. The first column is federal amounts, and the second column is New York amounts. The federal amounts on lines 1 through 19 should be the same as on your federal tax return, and include all your income. The New York column should show only the income that you earned while you were either living or working in New York.
If your New York income (while either living or working in New York) is less than your federal income, you are not paying New York tax on the full federal amount. The way New York calculates the tax, it first calculates tax on your entire federal income, after New York adjustments. But then it applies an "income percentage" that is your New York income as a percentage of your total income. That's how it adjusts the tax based on how much of your income is taxable by New York. You can see this on page 3 of Form IT-203, on lines 45 and 46.
If you have further questions, please say what the other state is that you lived in, and what state you worked in while living in each state.
