Is there a tweak that lets me delete any photos from my iPhone - including the ones uploaded with iTunes?
You can definitely do it on the phone, but if you have a lot of photos, that could take a while.
Assuming you are on a Mac...
Connect your iPhone to your Mac and open Image Capture (in your Applications folder). Click on the iPhone in the device list and it will start building a list of all of your photos. Once it's done, you can select the photos you want to delete (Cmd-A to select all) and then click the delete button at the bottom of the window (looks like a red circle with a slash through it).
If you don't care about the other data on your phone, you could erase the phone's data (including the pictures in those albums) by going to Settings > General > Reset > Erase All Content and Settings. Warning: all your data will be lost.
A friend of mine actually had the exact same problem a few weeks ago and ended up using this method.
When you are in the Photos app, simply press anywhere in the image you are viewing, and the toolbars will appear. On the far right of the bottom toolbar, there is a trash can icon. Simply press that, confirm you wish to delete the photo, and that's it. If photos keep coming back that you have deleted, then be sure you are not syncing an album that replaces those images.
Without using a computer, you have basically 3 options:
wipe your phone (as others have pointed).
jailbreak. then you'll at least be able to delete all the synced photos only. one way is trough iFile. go to
/private/var/mobile/Media/Photos/and delete it. The synced photos are in there but all "encoded" into databases, unlike the photos taken by the phone. There are ways to decode them, but I don't know of any on the iPhone.