PC troubleshooting help needed

Hey everyone, I’m looking for some help with my PC specs:
8GB RAM
Intel Core i5-8400
64-bit Windows 10 Pro
Whenever I try playing games, the computer just shuts down unexpectedly. The GPU fan suddenly kicks into high gear, even though the temperatures seem totally fine—around 37°C while running Need for Speed Heat.
I'm stumped because everything runs perfectly until I actually launch a game, then it just cuts out...
I've already tried a fresh OS install and reapplied the thermal paste, but no luck there.
I'm wondering if it might be the power supply, since it's a 500W 80 Plus unit, or maybe an issue with the electrical outlet?
If anyone has run into this before or knows what might be going on, I'd really appreciate the help...

What brand is the power supply?
What GPU are you running, or are you just using integrated graphics?

Pop the side panel off the case and see if that helps things at all...

Did you swap the thermal paste? Make sure everything was seated properly when you put it back together. Maybe too much paste?

If you're relying on integrated graphics, how much RAM did you allocate in the BIOS? You really need at least 1GB, preferably 2GB...