How do I back up my phone if I don't have iTunes?

You cannot. Even if you did have iTunes you couldn't backup the phone if the screen is unresponsive. Your best chance is to see if Apple can repair the screen without having to restore the device. If you were not using iCloud to do daily backups, you will only have the option of your last backup to restore the device.

jcOnnbb1214 Said:

"How do I back up my phone if I don't have iTunes?: What should I do if my screen is unresponsive?"

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Cant Find iTunes?:

iTunes is No More: For a more recent iPhone, the TV app is used for TV Shows and Movies, and the Music app is used for Music. Find Out More Here: What happened to iTunes? - Apple Support. So, if you do not see iTunes, then that is why, meaning you are all set.