If you chose to have your TurboTax fees deducted from your tax refund, refund processing won't slow down nor speed up your refund. Once Where's My Refund? shows "Refund Sent" it could take up to 5 days for it to be reflected in your bank account, it all depends on the bank, how and when funds are credited.
When you sign up for Refund Processing, your refund will be handled through the Santa Barbara Tax Products Group (SBTPG, formerly SBBT).
You can go to the Santa Barbara Tax Products Group website if you need assistance with your refund.