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.
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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.
If you efiled, download your return PDF and look at the page "Electronic Postmark." It will show the date submitted and the date accepted.
In did get the acceptance emails, but I don't see a transmitted date on any of the pdf tax forms that I printed.
You can check the transmission and whether your return was accepted on the IRS transcript website. To view your account transcript see the link below: