No you can not pay to get a faster refund. You might have paid the extra $40 Refund Processing Service charge to have the fees deducted from a federal refund. But that won't make it come faster. In fact it may add more days to it.
If your return was efiled and Accepted then only the IRS can tell you when. The date Turbo Tax gives you is just an estimated date of 21 days. The IRS doesn't update Turbo Tax about your return or refund.
What does the IRS say at Where’s My Refund
https://www.irs.gov/refunds