how can I unlock my child's iphone

If it was previously synced to a computer with iTunes, the "Erase your device with iTunes to remove your passcode" method described in the following link will make a backup as part of the process. If you forgot the passcode for your iPhone, iPad, or iPod touch, or your device is disabled - Apple Support

If it wasn't previously synced, there isn't a way to avoid losing data.

No.

And that is a good thing ... think how simple it would be to steal someone's data if an unlock could be performed without wiping the iPhone.

If you forgot the passcode for your iPhone, iPad, or iPod touch, or your device is disabled - Apple Support

No there is not.

No.

https://support.apple.com/en-us/HT204306