Q: My iPhone 5s won't let me download apps without verifying my credit card

Howdy Kat31413,

As I understand it you have a credit card on your iTunes account that you wish to remove. You should be able to do that with this article, either from an iOS device, or your computer:

Change or remove your payment information from your iTunes Store account (Apple ID)

If for some reason you cannot this article may help shed some light on why not:

Why can’t I select None when I edit my Apple ID payment information? 

Thank you for using Apple Support Communities.

All the best,

Sterling