iPhone turning off when locking phone after screen replacement

First things first, try to re-install the original screen to see if the problem goes away. Depending on where you bought the replacement screen and how you paid for it, you may just have a low quality screen.

If the problem is the same with the original screen, then you may have accidentally caused damage while you did the repair. Did you disconnect the battery prior to removing the screen?

I would look through this guide and double check my steps to insure you didn't do anything incorrectly. Also take the time to visually inspect the logic board, specifically the area near the connectors to see if there is any damage to the connectors or missing or partially dislodged components. Magnification will help here.