Turbotax doesn't pay your taxes. You pay your taxes, and sometimes you can pay through Turbotax.
Start by downloading or printing the PDF for "all forms and worksheets" from your account. This includes a page of instructions. For some states, Turbotax is not allowed to process electronic payments, and the instructions will tell you to mail a check or make a direct payment on the state's tax web site. Or, you may have accidentally selected "I will pay by check" in which case the instructions will say the same thing.
For other states, you can pay by automatic debit from your bank account. You would need to check your instruction page to see if you gave the correct bank account. Turbotax does not perform the withdrawal, it only passes the information on to the state. If you gave an incorrect bank account or routing number, or if your bank rejected the withdrawal attempt for some reason, then the payment would have failed. (You would also know this by checking your account statements.)
In some cases, Turbotax may offer credit card payment. In this case, Turbotax passes your information to a credit card processing company that pays your tax bill (and charges a service charge of 2-3% usually). Again, you should notice that no credit card charge was made.
Start by checking your filing instructions to see how you agreed to pay your tax, and then check the payment method (bank account, credit card). It's possible the state is wrong and a payment was made and they lost track of it. Or, you may have made no payment due to your own misunderstanding.
If after reviewing your return, you believe that Turbotax was at fault for not passing your payment information correctly to the state, you can request a refund of any late fees and penalties. Your case will be investigated. You owe the actual tax, of course.
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