Laptop is getting too hot?

I bought a new laptop a few days ago and installed a game on it (Genshin Impact) and when I play, the laptop gets very warm after a while and the fan also runs.

Can the laptop be damaged if i play too long or too often?

The device will be built to withstand prolonged heat.
That's what you've got the fan for. Is this good enough on?

Most devices have the fan BELOW, sometimes completely pointless because you put it somewhere. Take a look WHERE the hot air comes out and that it has enough space below, otherwise you will build something where you can put it on top, so that it still has enough space below (2-3 centimeters)

Actually, you can run a program over it to see the heat and then google whether this heat is normal for your device - I'm already typing.

Probably because the laptop is too bad.

Buy a laptop stand (there are also some with built-in fans) that works great.