In the PDF of your TurboTax return, the last page of the federal return should be an "electronic postmark" that shows your name, Social Security number, the date and time that you e-filed the return, and the date that the IRS accepted it.
Save the PDF
You can check the status of an Online efiled return here
https://shop.turbotax.intuit.com/efile/efile_status_lookup.jsp
There will be a date and time stamp for when it was received and accepted. I think there is also a Show Efile History button in the bottom left corner.
When you efile you get back 2 emails. The first email only confirms the transmission. The second email says if the IRS (or state) Accepted or Rejected your efile.
When you log into your account you should also see the status and if it was Accepted or Rejected, Started, Printed, Ready to Mail, etc. What does it say?
According to this post you need to call support to get the MEF or DCN number
https://ttlc.intuit.com/community/taxes/discussion/i-need-my-mef-number/00/2323037