Why is turbo tax saying my refund was approved and deposited but irs website still says processing

TurboTax did not say it was "approved."  TurboTax told you that your refund was "accepted."  Those do not mean the same.

https://ttlc.intuit.com/questions/2840013-does-accepted-mean-my-refund-is-approved

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 2022 Form 1040) of your federal refund to track your Federal refund:   

https://www.irs.gov/refunds

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

also Turbotax is never informed as to when a  refund is deposited. it uses an ESTIMATED date of 21 days after filing but the actual refund can come sooner or later or not at all if there are errors.