I had my fed taxes accepted a month ago and they are still not approved. No messages that something is wrong. What should I do?

There is nothing for you to do.  You cannot speed the IRS up.   You just have to wait for them to process your return.

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.  “21 days”  is not a promise from TurboTax or the IRS. 

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 2023 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://ttlc.intuit.com/questions/2580357-who-provides-the-refund-processing-service

https://www.irs.gov/refunds/tax-season-refund-frequently-asked-questions

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

Yes, you can track your refund using the Where's my refund? tool.

If that doesn't show any information yet, sign in to your TurboTax account and make sure your return status shows as Accepted. 

If you return is Accepted and the IRS hasn't contacted you, or posted any messages about delays in processing, there is nothing to do but wait.