iPhone randomly turns off but won't turn on

Check to make sure that you are doing the restart correctly.

Press and release the Up button

Press and release the Down button

Hold in the Side button and keep holding it in until the Apple logo appears on the screen. Ignore the option to power off the phone if that screen appears, just keep holding the Side button until the Apple icon appears, then release the Side button.

There are only 2 possible results from the restart......

1) The phone returns to normal operation

2) The phone has a hardware problem that will need the attention of the Apple Store or authorized Apple Service shop

See if this helps: If your iPhone won't turn on or is frozen – Apple Support (UK)