Can I use Turbo Tax to file 2016-2018 New York State Tax returns?

You can use TT IF you used TurboTax to prepare your federal returns for those years.  But if not, then no , you cannot.  Information flows from the federal return to the state return with this software.  It cannot be used to prepare a "stand alone" state return.

If you used TT for your federal returns but somehow neglected to file your NY returns for those years:

 Online preparation and e-filing for 2016-2018 is permanently closed.  You have to download desktop software for each year to a full PC or Mac (not to a mobile device) and then print, sign and mail your returns.  If you already paid for your TurboTax software, you can contact customer support to push a download of the software to you.

To call TurboTax customer support

https://ttlc.intuit.com/questions/1899263-what-is-the-turbotax-phone-number

They are available from 5 a.m. to 5 p.m. Pacific time.

To finish a prior year return started in TT online:https://ttlc.intuit.com/questions/2559403-how-do-i-finish-a-prior-year-return-that-i-started-in-turb

When you mail a tax return, you need to attach any documents showing tax withheld, such as your W-2’s or any 1099’s.  Use a mailing service that will track it, such as UPS or certified mail so you will know the IRS and/or your state received the return.