The fee you paid TurboTax (TT) to prepare a state return was only for preparation. There is a separate fee to e-file a state return (except NY). The free federal e-file does not include state. Check the "State filing instructions" work sheet that TT prepared (on your list of forms and worksheets) to see if your state return was e-filed or if you need to print and mail it.
If accepted by the state use this TurboTax support FAQ to check
the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund
