How do I reinstall iOS on my iPhone without losing data?

When you make backup, right click your iPhone in your iTunes’ sidebar, then select “Restore from Backup”. iTunes should ask you now what backup you want to restore from, just select the latest one (should be preselected) and it should start to recover lost data from iPhone.

Can you restore your iPhone without erasing everything?

We’re happy to help. Restoring your iPhone removes the content that is on your phone. It will not delete the content you backed up to iCloud.