So the games run fine, this would mean your drivers
are working ok and, more than likely, your issue is with heat, more so when the temps are that high just browsing on the internet... have you check how hot it gets before the screens shuts off?
Not much you can do on a laptop unless you want to spend some money on a laptop cooler... even then, I don't believe it will be enough.
Are you playing with the laptop in, well, your lap? if so, trying playing it on a surface that does not get to hot underneath.