TurboTax gives you an estimated date for receiving your refund based on a 21 day average from your date of acceptance, but it can take longer.
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.
You need your filing status, your Social Security number and the exact amount (line 20a of your Form 1040) of your federal refund to track your Federal refund:
To track your state refund:
On the IRS site you are also offered the use of an app called IRS2Go
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.