How do I tell when my return has been officially accepted if I sent it by mail? I do not mean tracking my mail, I know it has arrived already.

Just keep checking the IRS website "Where's My Refund."  It may take several weeks to show up there.

TurboTax never knows that your put your tax return in an envelope and took it to a mailbox, so there will be no information on the TT site.  When the IRS opens your envelope and puts the information into the system you will be able to track it on the IRS refund site.   That is your only source of information.

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:   

https://www.irs.gov/refunds