How do i backup my iphone if the iphone itself wont turn on?

Things you can do:

• Try to replace just the screen and see if it works;
• Leave the phone in repair shop and pray that they won't wipe it during repair.

Not much else you can do, sorry.

The time for backups is before you need them. Apple makes it trivially easy to set up the phone to back up automatically every night, but you still have to turn that feature on.

If the only damage is to the screen Apple can replace the screen without erasing the phone most of the time.

You can’t back it up now. Does the phone have actual physical damage? If so, it sounds like you may need a new phone.

Nothing you can do until you get the device repaired.