iPhone 13 battery expansion causing screen damage: Can Apple repair or replace?

First: Don’t charge that phone again!

Second: Don’t leave it unattended and resting on flammable materials.

It CAN get worse. (e.g. Fire)

In all likelihood, your battery needed replacing long before it completely failed and physically deformed as you describe.

Unfortunately, it’s probably time to buy a complete replacement phone as the swelling likely caused other internal damage.

Apple Service and Repair for iPhone - Apple Support

Is there something I can do to get apple to repair/replace mine.

Replace will likely be your only option. From your description, if the swelling battery pushed out the display, surely other internal parts are destroyed or at least seriously compromised.

NOTE: You cannot ship a device with a swollen battery. You will have to hand-carry it for evaluation or recycling. At this point it is flammable hazardous waste.

Sorry, but Chattnoogan is 100% correct. Your battery was likely failing and there was surely a notification in settings that Battery Service was required. You surely ignored the notification and it's highly unlikely Apple can fix your phone now. Do schedule an appointment at your Apple Store Genius Bar and they can tell you if the phone can be fixed, but be prepared to buy a new iPhone. I also agree the phone should NOT be used or charged and if possible placed in a metal container powered off until you take it to the Genius Bar at your Apple Store --> Genius Bar Reservation and Apple Support Options - Apple