Religion is a collection of beliefs, cultural systems, and moral values. No matter what the belief is, it falls under this category, whether Atheism(yes, believing there is no God is a belief system), Buddhist, Christian, Muslim, or other.
Science, on the other hand, is a system of building and organizing information. It proves this information by experimentation.
You'll notice that neither definition has anything stating that the other is false. Of course, I didn't go into detail regarding individual religious beliefs, that would take a very long time. Regardless of this, most religions only disprove each other, not science. I have disregarded some minor sects that try to contradict science, such as dadaism. I don't count as a regular religion, however, as they were created for the sole purpose of fighting logic and reason.
Now to the heart of the matter. Science is being declared to disprove religion; most commonly through the theory of evolution. First off, that is only a theory. It requires just as much faith as the most convoluted cult. It does have a basis in fact. Micro-evolution is easily true. Macro-evolution on the other hand has yet to be concretely proven. If something does not have concrete proof, it falls under the belief category(skip to the last paragraph of this post for an explanation of macro and micro evolution). I can find just as much truth in any other religion. This is because every religion is based off of the beliefs that someone came up with after observing fact.
Therefore, science does not contradict nor disprove religion. It actually compliments it in most places. Neither does religion shackle science; if nothing else, it gives it purpose.
Macro and Micro Evolution
I'll make this quick. Micro evolution involves relatively minor genetic changes, such as a mutation, to change creature into another, similar creature; for example, a regular carrier pigeon becoming a separate and distinct species of pigeon. Macro evolution states that if one pigeon can become another, then it is reasonable that a pigeon could become an eagle, given enough time. Sure, that is reasonable, but it has not been proven.