If you e-filed the tax return using TurboTax you would have received two emails. The first email was when the tax return was transmitted and will have a date/time included. The second email would be when the IRS or State either Accepted or Rejected the e-filed tax return.
There is also an Electronic Postmark Report included in the PDF download with will also have the dates and times for the federal return that was e-filed.
If the tax return was printed and mailed, TurboTax has no way to know when the user mailed the return.