If I worked in New York for a short term internship as a nonresident, don't I only have to pay NY taxes on the income I earned there?

It depends -

According to the NY Department of Revenue -

If you are a New York State nonresident you must file Form IT-203, Nonresident and Part-Year Resident Income Tax Return, if you meet any of the following conditions:

• You are a nonresident with New York source income and your New York adjusted gross income Federal amount column (Form IT-203, line 31) exceeds your New York standard deduction.

So if you are a US citizen, you are filing a US income tax return, you have NY sourced income as a nonresident of NY and meet the NY state filing requirement, then you do need to file an NY nonresident state income tax return.

You can add a return for NY under the state taxes tab.

Just follow the TurboTax guide when working on your states (remembering to do your nonresident state return first) and TurboTax will do all the calculations and credits to your resident states return.

Here is additional information about filing in multiple states (select "see more answer" to view the entire attachment)

https://ttlc.intuit.com/replies/3300797