TurboTax has no way at all to track a tax return that you mailed.   The software does not know that you put your return in an envelope and took it to a mailbox.  Anything you are seeing as "accepted" can only refer to a return that you e-filed---not one that you mailed.    So that is referring to the original return that you e-filed---not an amended return that you mailed.   

And you are correct--mailed returns do take a long time to be processed, especially amended returns, which take 4-6 months.     Eventually you will be able to track it on the IRS amended return site:

https://www.irs.gov/Filing/Individuals/Amended-Returns-(Form-1040-X)/Wheres-My-Amended-Return-1

https://www.irs.gov/filing/individuals/amended-returns-form-1040-x/amended-return-frequently-asked-q...

Thanks for your answer.   I remember when I started my amended return, using a copy of my already e-filed return,  it said accepted.  Of course, that's where the status was pulled from.

Also, your link Where's my Amended Return worked as opposed to the Where's my Refund.  It shows it has been accepted at that link.