You didn't say whether you are a New York resident or a nonresident. I will assume you are a resident, filing Form IT-201. If that's not correct, disregard the rest of this reply.
It's not clear what you mean by "according to NY.gov." I don't think you can just pull your tax off the state web site. At your income level you have to calculate your New York tax using one of the worksheets under "Tax computation—New York adjusted gross income of more than $107,650" starting on page 45 of the Instructions for Form IT-201.
Also make sure you are comparing apples to apples. I'm not sure what "tax owed" you are looking at in TurboTax. When you calculate the tax on Form IT-201 line 39 using the appropriate worksheet, compare it to IT-201 line 39 in TurboTax. Look at the actual tax forms, not a summary screen in TurboTax.
