When you filed your return you would have received an email that told you that the return was accepted. "Accepted" only means you had enough data for the IRS to take it. They still have to put it through a longer process before the refund is approved and issued.
https://ttlc.intuit.com/questions/2840013-does-accepted-mean-my-refund-is-approved
Take the words "still being processed and a refund date will be provided at a later date" literally. The IRS is not finished with your tax return. It is being checked for mistakes, etc.
Once your federal return has been accepted by the IRS, only the IRS has any control. TurboTax does not receive any updates from the IRS. Your ONLY source of information about your refund now is the IRS.
You need your filing status, your Social Security number and the exact amount (line 35a of your 2021 Form 1040) of your federal refund to track your Federal refund:
To track your state refund:
https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund
https://ttlc.intuit.com/questions/1901548-why-do-some-refunds-take-longer-than-others
If you chose to have your TurboTax fees deducted from your federal refund, that will take some extra time, while the third party bank handles the refund processing.
https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions