"How can I get a document that verifies I got an extension"
If I understand the thrust of your question, you don't. The IRS does not send you a confirmation that you received an extension; the presumption is that if you created and filed form 4868 correctly and in a timely manner that you received the extension.
This is one reason that taxpayers are encouraged to e-file when possible or if mailing, to retain at least a "proof of mailing" (not the same as proof of delivery), to show that you mailed something to the IRS on that date (you can handwrite on the receipt what the mailing was for).
If you filed the extension through TurboTax, follow xmasbaby0's advice above and print off the 4868. I don't know if TurboTax has a screen that states that you filed the extension when you re-enter the return a second time ( @xmasbaby0 do you know?), if so, screen print a copy of that, too, and add it to the 4868 for your "proof of extension".
