Tax file

If you mailed it then you are the only one who knows when you took it to a mailbox, unless you saved a receipt from using certified mail or UPS, etc.

If you e-filed it, you got emails after you e-filed:

When you e-file your federal return you will receive two emails from TurboTax.  The first one will say that your return was submitted.  The second email will tell you if your federal return was accepted or rejected.  If you e-filed a state return, there will be a third email to tell you if the state accepted or rejected your state return.

And your account should tell you what date you e-filed--if it was accepted.