There are a lot of different aspects to relativity and scientists are still designing experiments to test its implciations. However, there is a great deal of relativity that has been tested enough to be accepted as proven. . . for example time dilation, tested by measuring the difference between a clock in a fast plane and a ground clock.
Why is Conservapedia is against relativity? Good question. Maybe you find which editors wrote the article, and then find their personal views.
My guess is that we are witnessing another episode in the age old conflict between power, religion, and science. Political leaders who have been close to religion have continually, throughout history, been in conflict with science because science presents an alternative source of power in society. The process of science promotes people to question authority, and rewards the replacement of the old with the new. This is a problem for the religious political leaders of any society, who are trying to maintain order and balance amongst competing political interests, to prevent violent rebellions, to keep the 'status quo', which their benefactors expect them to do, and which they see as some kind of moral duty.
I would suggest several sources to learn more about this history. The first and best and easiest to deal with might be the television program "Cosmos", by Carl Sagan et al, from the 1980s, available on Hulu.com for free viewing. He discusses some of these conflicts through history and tries to explain why they happened.
Another source would be to study the ancient Hindu astronomers and mathematicians - they too had problems with the religious authorities. That is why the mainstream of Hindu astronomy did not really accept a rotating Earth for a long time - because it would have contradicted statements in the ancient Vedas. To challenge the Vedas was to challenge the power of priests in Hindu society - a very dangerous thing to do.
Then you can look at the history of Galileo and how the Catholic church tortured him for his work. The conflict was not really about the relationship between the sun and the earth - the Church really did not care. The conflict was about power and the way people should view the world. Galileo, of course, represented science - the questioning of the past and the writing of a new future, and specifically the questioning of the bible and the Church authorities. If you could challenge them on the Sun and the Earth, then what about things like priests abusing their power, taking corrupt bribes from rich people, allowing 'indulgences', stealing land, and on and on and on?
Then you can come to Evolution. Darwin was quite worried about the conflict between Evolution and his dear religion, which was why he almost did not publish his theories. But eventually he did. And then we had the conflicts between the religious political leaders and evolution and science. It is not that they actually care where the Earth came from - what they care about is power and order in society. If the populous can start being educated to question authority, then there is no telling where it will end. It is hard to field a vast military or run a huge corporate work force or an imperial bureaucracy (the UK) if you have a bunch of people educated with the scientific mindset - to always question why things are done and to always try to test existing theories. For these leaders, blind obedience to the church, the state, or the organization, is vastly more important than adopting some kind of scientific world view.
Now there is relativity. Einstein, then, basically, represents the scientific world view... the questioning of authority, and the challenging of power.
Einstein himself was somewhat of a socialist, definitely a vegetarian, and also a pacifist. Einstein was not allowed to work on the Atomic bomb project because he was deemed politically unreliable. Einstein was not trusted with security clearances. Einstein with all his fame was essentially viewed by the state authorities of the United States (and Nazi Germany, where he was from) as an 'enemy' not to be trusted. He was part of the group of scientists in the 20th century, like Andrei Sakharov, Carl Sagan, and many others who began a sort of massive pacifist anti-state movement against nuclear war and against the mis-use of science in general by governments and state authorities. They formed groups like the Federation of American Scientists (fas.org) and they put out the Bulletin of the Atomic Scientists and created the idea of the 'doomsday clock' to alert the population of the danger of nuclear war. 'Conservatives', whether in the USSR or US, did not trust these people and did not like them. From their perspective, pacifism was a dangerous idea and probably enemy propaganda planted by the other sides intelligence organization (and sometimes this was accurate).