You cannot pay to have the IRS process your tax return and send the tax refund faster.
The TurboTax Free online edition is for very simple tax returns that do not require any other form or schedule other than the Form 1040. If your tax data requires entry on any other form or schedule then you must upgrade to the Deluxe edition or higher.
Go to this TurboTax website for the Free online edition - https://turbotax.intuit.com/personal-taxes/online/free-edition.jsp
Only the IRS and your State control when and if a Federal or State tax refund is Approved and Issued.
You complete your tax return by finishing all 3 Steps in the File section. In Step 3, to e-file your tax return, you must click on the large button labeled "Transmit my returns now".
After completing the File section and e-filing your tax return you will receive two emails from TurboTax. The first email when your tax return was transmitted and the second email when the tax return has either been accepted or rejected.
To check the status of an e-filed return, open up your desktop product or log into your TurboTax Online Account. You can find your status within the TurboTax product.
If accepted by the IRS use the federal tax refund website to check the refund status - https://www.irs.gov/refunds
If accepted by the state use this TurboTax support FAQ to check the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund
