State Refund

That cannot be correct.   TurboTax fees can never be paid from a state refund.   Never.   Only a federal tax refund can be used to pay TurboTax fees---when YOU choose "refund processing" and allow a third party bank to receive your federal refund, take out the fees owed to TurboTax, collect the bank's own fee of $40 ($45 in CA) and send you the rest of the federal refund.

The third party bank is not involved in any way with your state refund.  Your state refund comes to you directly from your state.  If you owe back taxes or child support, (or some other kinds of state debts)    your state can seize your state refund and will send you a letter about that.

Federal and state refunds come from completely separate entities,  There is no rule as to which one will come first or how much time there will be between their arrivals.

STATE RETURN

Make sure your state return was accepted:  https://turbotax.intuit.com/tax-tools/efile-status-lookup/

To track your state refund:    

https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund

https://ttlc.intuit.com/community/state-taxes/help/how-do-i-contact-my-state-department-of-revenue/00/25573

How can I see my TurboTax  fees?   

https://ttlc.intuit.com/questions/2565973-how-do-i-review-my-fees-in-turbotax-online

What is Refund Processing Service?

Santa Barbara Tax Products Group, LLC  (SBTPG) is the bank that handles the Refund Processing Service when you choose to have your TurboTax fees deducted from your refund. This option also has an additional charge from the bank that processes the transaction. 

You can contact them SBTPG, toll-free, at 1-877-908-7228 or go to their secure website www.sbtpg.com 

https://ttlc.intuit.com/questions/2580357-who-provides-the-refund-processing-service