You can file with multiple states in TurboTax.
Start with the Federal return and enter all the W2s including the Lines 15 through 17 for all the states. If you just worked in the other states you will pay taxes in those states and get a credit for taxes paid in your home state. You should do the non-residential states first and your residential state last. This is to ensure you get the credits derived from the non-residential states to flow properly.
Check to see if any of the states have reciprocity with your home state. If you do have states with reciprocity check to see if that state withheld on your W2. If the state withheld and they do have reciprocity with your home state you will need to file a return to reclaim the amount withheld. If they did not withhold then you are good to go.
[Edited 03/14/2020 08:37 PDT]
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