You can e-file by going through all of the steps in the FILE section of your TurboTax software program---you need to enter banking information for direct deposit of your refund if you want that. And you go all the way to a big orange button that says "Transmit my returns now" and click that button to e-file your return. That sends your return electronically---but it is not "e-mailing" it.
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.
First, check your e-file status to see if your return was accepted:
https://turbotax.intuit.com/tax-tools/efile-status-lookup/