Late filing for 2015

Turbotax Online works only for current year (2016) taxes, and not for any other year.
If you want to use TurboTax for any other year you will need to purchase the Desktop Version for each year that you need to file. You can do that at this link:
http://shop.turbotax.intuit.com/personal-taxes/past-years-products.jsp

You can download the Forms and Instructions free from the IRS web site at this link:
http://www.irs.gov/app/picklist/list/priorFormPublication.html

In any event, you have to file the return by mail. Turbotax does not support e-file of prior year returns. Tip: If you prepare multiple years of prior returns, mail each one to the IRS in a separate envelope to be sure that each one is recognized as a separate tax return.

I have 3 past years late returns to mail (2014, 2015, 2016).  As long as  I mail each of them in a separate envelope, will it cause a problem if I mail all of them in the same day?

That should not be a problem.  But do put them each in their own envelope.  When you mail tax returns it is a good idea to use a mailing service that will track them, like UPS or certified mail so you will know the IRS received them.