I have had a Kindle Fire for approx. 2 years. When reading the Daily Mail page, after a couple of mins, it goes off itself or turns the screen white or just freezes totally. All other sites seem OK. Please don't say stop reading DM, I like the gossipy stuff!
I don't know if this will help but the mail on my Samsung tablet freezes from time to time and I found that if I go into the app settings and clear the cache it works again ok, maybe its the same on your Kindle?