iPhone 5S will not update or download anything

I noticed maybe a month ago that my iPhone had stopped saving web history and stopped allowing me to download apps, updates, software updates, or music. This was not a problem until one of my more important apps released an update and no longer supports the old version, leaving me unable to use it. I have tried resetting my phone, deleting apps and photos for storage, logging in and out of my Apple ID but nothing seems to work!