When I filed, I entered in tax payment information, but I can see no way to track that my payment went through. That's making me nervous. I don't want a late fee.

Next time, pay directly on the IRS website, and you'll get an instant confirmation.  But also see this:

IRS Computers Are Behind On Debiting Scheduled Payments (forbes.com)

Understood on both counts. Disappointing that TurboTax provides no means of tracking what I have already done, however. Thank you.