How to stop the iPhone screen turning on when receiving a notification?

I still want it to vibrate and make a sound, but it's annoying when the screen turns on by itself!

Go to settings and select push notifications then turn off notifications of the apps you want to

I like the fact it turns the display so I could take a quick glance & act if necessary instead of an Android phone that does what you want it to do in this case, but then I would have to act either way to find out what it was that I was notified about.

Anyway, I'm not 100% sure about this since I've never done it, but you could turn off lock screen notifications for apps in the settings. Again, I'm not sure if the phone would still vibrate & alert you to it and/or light up, but it is an option you could try.

Turn on do not disturb. You get text messages while your screen is off. And when wake up the iphone the text messages will be in the lock screen.