Fans only work in overdrive if they need to. If all your case fans are going crazy the overall temperature in your case is too high. Check the temperature in BIOS if you know how, it shouldn't be above 50 degrees celcius (60 Max). That could also have an effect on the CPU, not sure why it's only for certain websites. I assume you are using a browser like Chrome or Firefox?
I would definitely recommend cleaning the CPU properly, even get some moisture-free air in a can and blow all the dust away. Vaccuum cleaners work well too.