Just a guess but it sounds like your fan speeds weren't increasing when the CPU got hotter and the high temps shut it down. Checking the box forced them to run faster/cool down better in sync with temps. Still odd that it was only those distance sites that did it since they don't really look that resource intensive to me. Maybe they have busy banner ads or something. I use an ad blocker so don't see that crap.

What temps are/were you seeing?

Normally fan speeds are preset and controlled pre-boot in the BIOS or UEFI in newer PC's esp if they're a brand name computer out of the box. Those may have been changed at some point after purchase or perhaps just corrupted somehow. What is the model/make of the computer? Or the model of the motherboard if you know it?