Why does my iPhone 5c say it cannot take new photos even after I delete 100 or so?
Restore from a backup.

It looks like there are a lot of cached files on your phone that shouldn't be there. Usually restoring from a backup will clear all of this system trash as it only backs up personal information.

Note: an iTunes backup will be more complete and will contain a lot more app data than iCloud, so that is recommended if you are able to use iTunes.

iTunes (recommended)


Make an iTunes backup. Connect your phone to your computer, and open iTunes. Select Your iPhone (in the top bar) - Summary - Back Up Now.
Disable Find My iPhone. Go to Settings - Apple ID - iCloud - Find My iPhone and turn off the switch. You will have to enter your Apple ID password. 


If you have forgotten your Apple ID password, go to iForgot

In iTunes, click Restore iPhone.
Wait for the restore to complete
In Setup Assistant, select that you'd like to restore from an iTunes backup and follow the directions from there.




iCloud


Make an iCloud Backup. Go to Settings - Apple ID (big thing with your name at the top) - iCloud - iCloud Backup - Back Up Now
Disable Find My iPhone. Go to Settings - Apple ID - iCloud - Find My iPhone and turn off the switch. You will have to enter your Apple ID password. 


If you have forgotten your Apple ID password, go to iForgot

Wipe your phone. Go to Settings - General - Reset - Erase All Content and Settings.
Wait for your phone to do a full wipe.
In Setup Assistant, select that you'd like to restore from an iCloud backup. Choose your desired iCloud backup.


That's it! See if it worked.