Iphone 5 stuck in boot loop

Press Sleep/Wake and Home buttons and hold them for at least 10 seconds.

If this does not help, take a look these Apple Support articles:

If your iPhone, iPad, or iPod touch doesn't respond or doesn't turn on - Apple Support

Use iTunes to restore your iPhone, iPad, or iPod to factory settings - Apple Support.

As a final step, try the recovery mode. Turn your device off, then plug it to your computer with the home button pressed. Keep the Home button pressed until you see the iTunes logo on the screen of your iPhone. After that on your computer you should see the iTunes window saying your iPhone needs to be restored to factory settings. Click on the restore button.

More info here: https://support.apple.com/en-us/HT201263

If these steps don't work, contact Apple and ask for help.

https://www.apple.com/support/iphone/contact/

What is the error code?

Take a look at these articles:

Resolve iOS update and restore errors in iTunes - Apple Support

Get help with iOS update and restore errors - Apple Support

Below you will find the troubleshooting steps suggested by Apple:

Check for hardware issues

Errors: 1, 3, 11, 12, 13, 14, 16, 20, 21, 23, 26, 27, 28, 29, 34, 35, 36, 37, 40, 53, 56, 1002, 1004, 1011, 1012, 1014, 1667, or 1669.

Your device or computer might have a hardware issue that stops the update or restore from completing.

1. Check that your security software and settings aren't blocking your device from communicating with the Apple update server.
2. Then try to restore your iOS device two more times while connected with a cable, computer, and network you know are good.
3. Make sure that your security software and settings are allowing communication between your device and update servers.
4. If you still see the error message when you update or restore, contact Apple support.

If you still see the error message, check for outdated or modified software.

Nobody can tell you this for sure without examining the phone. Take it to the place where you got the screen replaced and ask for help.