A state tax due can only come out of your bank account or you have to mail in a separate check. At the end Turbo Tax just shows you the net combined amount of federal and state but that is only for your info.
Check the printout or PDF of your return; look for the state cover sheet with the Turbotax logo. If you owed tax, it will show the payment information and how/when you decided to pay. Read the state payment instructions carefully since most states cannot be paid from within the TT system and requires additional steps.