I'm 19 years old and worked at a camp in NY summer 2019; live in PA. Turbo Tax says I owe NY taxes on my entire 2019 income, chose nonresident for NY. How do I fix this?

Tax for NY nonresidents is computed as if the taxpayer were a full-year resident of New York, then the tax is multiplied by an income percentage based on New York-source income and total federal income.

The income percentage should show up on the summary page for your NY taxes.  If that income percentage is not accurate, you should review the entries in your New York (NY) nonresident return to make sure that only the NY income is designated as NY income.  When it is properly entered, your "income percentage" will reduce the total NY tax to only the part related to your NY income. 

Note that you can get a Pennsylvania credit for taxes paid to another state on the "same dollar of income".  TurboTax will have a section or question related to that credit.