Screen is doing weird things without me touching it

First do Forced restart

https://support.apple.com/guide/iphone/force-restart-iphone-iph8903c3ee6/ios

If the same thing happens you may have a hardware problem and the  next procedure would be to backup then restore your device to factory, and setup the device without restoring a backup, to see if the issue is replicated. 

Then restore the backup. 

If you need to send the phone in for repair then backing up and restoring to factory would be needed anyway.

Here is a guide for those steps

Backup and restore

(1)How to back up your iPhone, iPad, and iPod touch – Apple Support

Remove SIM PIN if you have set one

(2) Restore your iPhone, iPad, or iPod to factory settings – Apple Support

Set up as new from the Hello Screen without restoring backup. Test if the problem still exists

If the problem goes away in the newly set up phone then suspect settings or apps, and in particular VPNs and so-called security apps which should never go near an iphone.

If the problem remains then arrange a repair including details of these tests.

https://support.apple.com/en-gb/iphone/repair/service

(3) When your phone comes back Restore your iPhone, iPad, or iPod touch from a backup

You should arrange a repair

https://support.apple.com/iphone/repair/service

Anything other than Screen or Battery may be a replacement phone and the price is in the article, depending on your warranty or AppleCare+ status.

You can see additional service information within the following article: 

Get your iPhone, iPad, or iPod touch ready for service - Apple Support