Is going full screen, while watching YouTube, causing my Android Motorola phone to crash?

Yes. Although it wasn’t necessarily Youtube – it could have been anything.

Phones, like any computers, have microprocessors (CPUs). When they are working (computing) they produce heat. Most newer phones have a sensor internally that will turn off the phone if the CPU temperature reaches some limit – so as not to damage the CPU or any other components of the phone.

Watching a video, full screen, is using a lot of CPU and therefore creating a lot of heat.

So in this case, the protective circuits inside the phone did exactly what they are supposed to.

I have a Moto G Power and so far no problems. :)