How will I know when my mailed out tax return was received?

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"?

The only way to know that the IRS received a return you sent in the mail would have been to mail it with a service that tracks it like certified mail.   If you did not do that, then all you can do is wait for 3- 4 weeks and hope it shows up on the IRS refund site when a human puts it into the system.

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