If you set up the payment to happen by direct pay why did you also pay a second time? When the fed or state finally gets around to processing your return and they see the excess payment they will refund it automatically and there is nothing TT or you can really do to speed up the process.
If you set up a payment through Turbotax, that information was passed on to the IRS or the state. Turbotax doesn't have your money and can't make the IRS issue a refund if you paid twice. If you only set up payment in Turbotax, but the IRS took 2 payments, that would be unusual, but also an IRS mistake, and Turbotax still doesn't have your money and can't make the IRS issue a refund.
You can verify that the IRS did receive 2 payments by checking your Account Transcript here.
However, it may take 4 weeks for recent account activity to show up on your transcript.
If it is more than 4 weeks since the second payment, and both payments are still not listed on your account transcript, you can contact the taxpayer advocate service to initiate a trace of the payments and make sure that both payments are credited to your account. The TDD number for the IRS is 800-829-4059.
Once both payments are registered as showing on your account, the IRS will recognize that you overpaid, and will refund the extra payment. This may take a few more weeks.