I had a look at the page you linked, and couldn't see a single peer reviewed study. Just lots of links to other webpages which led me down quite the merry path. Now the internet is a big place and research is unfortunately not immune to the influence of politics or economics and one of the great thing about facts is (as Stuart Lee said) "You can use facts to prove anything"
The WHO seem to be quite sure that mentions of palm oil as a health risk are inaccurate http://www.who.int/dietphysicalactivity/media/en/gsfao_cmo_068.pdf
Also the World Journal of Cardiology wrote the following piece as well
Note that in the animal experiments the rats taking the oil were compared to a group that took no oil, not a group that took the same amount of a different oil, so of course there would be some adverse affects. If there were various controls groups having their diets supplemented with sunflower oil, flax oil, corn oil, cod liver oil, or even Total motor oil etc then we might see results that were specific to palm oil as opposed to others
I'm not saying that palm oil is good for you but I'm also not seeing studies that legitimately single out palm oil as being particularly worse than any others. Obviously be careful about what you put into your body, and doing research is always a good thing, however headlines can be misleading, and as I said earlier you can use facts to prove almost anything