Nokia 5.3 screen turns black on making call.

This type of screen blackout problem was reported in Nokia mobile earlier versions also. In fact it is a common problem in many earlier series. Right now there is no easy solution for this problem but there are some measures which were taken in its earlier versions and the user got this problem resolved. So one method is to restart the device. There will be some menu when you press the main switch button for a longer time and in that menu just find out the restart option. Click on it and the device would start afresh and the problem will be resolved. In case if you do not get success then I would suggest you to go to Nokia users forum in the web where you can get a lot of information about these glitches and their solutions. One popular site for all these types of information is https://community.phones.nokia.com/discussion where you can find the relevant information shared by Nokia users world wide.

The most probable reason may be the proximity sensor on your phone is not working properly. The proximity sensor is a little sensor, positioned generally on the top of the phone. The sensor will recognize when the phone is near your ear. The phone will get locked and then you won't accidentally press keys with your cheek. After you finished the conversation and remove the phone from your ear the sensor will unlock your screen and allow you to end the call.
You have to test the proximity switch on your device. I suggest you go to the service centre and tell him the problem, He will have some codes to test the proximity switch and fix the problem. If you know the codes you can verify your selves.
If your sensor is clean and no problem, the other reasons may be like a bug, glitch, or virus on the phone. You take a full backup on your phone, save your pictures, contacts, text messages etc. Then perform a full wipe on the phone. This usually fixes your problem. Even then your problem is not solved, you have to give it back to the company and get a new one as a replacement if it is in guarantee period.