Background Games

Maybe it works for you, you can open spotify and play music in the background while you play, a good list of lectro and forzahorizon is a good option or something of metal and gow 5 surely gives you a more aggressive experience

The app download games in Background but when you playing any game the speed connection to download background is less

It may just be some sort of processing going on with your Tv, if you could provide the Tv make/model we may be able to help regarding disabling these settings.

Have you tried using a different cable to plug into your tv from the Xbox. If you are using the same HDMI cable, I suggest changing it out.

What game are you referring to?

Make sure your display setting are correct in your console settings (widescreen, 720p, etc.).

Try playing the game on your parents' TV and see if it still occurs, if it does, it could possibly be the game itself, the settings you're using, your cables, or (unlikely since it was occurring on your old XBOX as well) your XBOX.