The evidence therefore seems to suggest the problem is with either the wifi hardware or software driver on the laptop. I'd be more confident of that if you had another laptop rather than smartphones for the test. Not sure why it would be only on some websites though.

I guess it could also be some interference with the wifi signal, but you would think that would affect the phones too. Have you tried moving the laptop to right next to the router?

Anyway, you could try a usb wireless adapter, to eliminate the built in wifi hardware and its driver. They are cheap, like this one