Why does TurboTax say my refund is accepted, but the IRS website says it has not been approved yet?

Accepted only means the IRS received the tax return without errors on the return.

After the tax return has been Accepted by the IRS (meaning only that they received the return) it will be in the Processing mode until the tax refund has been Approved and then an Issue Date will be available on the IRS website.
See this IRS website for federal tax refund FAQ's - https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions

Once your return is filed, TurboTax has no control over the processing of your return or the release of your refund.

Here are a couple FAQs to track your refund:
When will I get my federal tax refund?
How do I track my state refund?
Why does Where's My Refund say "approved" but TurboTax still says "accepted"?

Why do some refunds take longer than others?

What if my IRS refund is taking longer than 21 days?

  “Accepted” is not the same as “approved”.  TurboTax tells you the e-file was accepted if the IRS deems that there is enough information on the return for them to take it in for processing.   Only the IRS can approve of the refund, which is a later stage of processing.  If the IRS approves your refund they will provide a date for the refund to be issued.

First, check your e-file status to see if your return was accepted: 

https://turbotax.intuit.com/tax-tools/efile-status-lookup/

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 2024 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/turbotax-support/en-us/help-article/tax-refund/track-state-refund/L3jgO8PGs_US_en_US?uid=lt447ebr

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://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-refund/refunds-take-longer-others/L14YlqFrH_US_en_US?uid=lexdr7zh

.

https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-refund/irs-refund-taking-longer-21-days/L2vRAJbdU_US_en_US?uid=lexe7lst

If you are getting earned income credit on line 27 or additional child tax credit on line 28 You are subject to the delay required by the PATH act.  Do not expect your refund before early March

https://turbotax.intuit.com/tax-tips/irs-tax-return/what-is-the-path-act-and-how-may-it-impact-your-taxes/L2fkZJmNC