How to fix my iPhone 13 black screen of death?

Try a Forced Restart on the phone. Follow the instructions very carefully, as this is a bit tricky. (Hint.....ignore the "slider" when it appears and keep holding the Side button in until the Apple logo appears. This might take another 30 seconds.)

Force restart iPhone - Apple Support

There are only 2 possible results when you perform a Forced Restart:

1) The phone returns to normal operation

2) The phone still won't start up.....because there is a hardware problem with the phone that will need attention at the Apple Store (make an appointment) or take the phone to an authorized Apple Service shop.

You can find a list of authorized service shops in your area using the link just below:

Find Locations

Hello,

Start here:

1. Press and quickly release the volume up button.
2. Press and quickly release the volume down button.
3. Press and hold the side button until you see the Apple logo (could be as long as 30 seconds) 

The timing is a little critical, you may have to try it several times. See this for more detail:

If your iPhone, iPad, or iPod touch won‘t turn on or is frozen

Good luck.