There are a few different ways that you can back up your iPhone if the screen doesn’t work. You can use iTunes on a computer, iCloud, or a third-party app.
If you have a computer and iTunes installed, you can plug your iPhone into the computer and open iTunes. If your iPhone is unlocked, it will sync with iTunes. If your iPhone is locked, you will need to enter the passcode to sync.
If you don’t have a computer, or …
