Are religion and science compatible?

Factually maybe, but philosophically no.

For fundamentalists, religion and science are just plain incompatible on all levels. The findings of geology, geometry, astronomy & history directly conflict things in the Bible, and fundamentalists believe the Bible to be literally true and inerrant. Since they believe a bronze age manuscript to be the ultimate source of knowledge, they reject the findings of science.

Most non-fundamentalists would probably say that religion and science are perfectly compatible. There is no problem reconciling (say) liberal or mainline Christianity or Islam with our knowledge of chemistry, physics, biology or pretty much any branch of the physical and social sciences. The knowledge base of science, the facts of science, are not in conflict with an allegorical reading of the Bible.

However, I think there is an enormous conflict between the philosophies of science and religion.

Science involves a mix of creativity and skepticism. Ideas and hypotheses about reality can come from anywhere - dreams, hallucinations, insights, plucked from someone's ass or even from ancient manuscripts. But the skeptical half of science requires thoroughly testing those ideas against reality. Ideas that don't fit with what we observe are ruthlessly discarded. In this way the gold is separated from the muck and we can find nuggets of truth about the universe. No individual scientist is perfect, nobody is correct all the time, but Science as a whole is an error-correcting, evidence-based process for finding out what is really going on. So far, this is the only reliable method humanity has figured out for discovering truth.

Religion is based on faith. Faith is belief without evidence (or even in spite of the evidence). While it may happen that the proposition believed is in fact true, this is accidental. Religion has no method to find out which of its beliefs are true and which aren't. If there happens to be evidence, religous people will use it, but if not, the need for evidence is dismissed. Doubting Thomas is not held up as a role model of honest intellectual inquiry, but as an example of what not to do. It is better to not doubt or require evidence, as Jesus makes clear: "Jesus saith unto him, Thomas, because thou hast seen me, thou hast believed: blessed are they that have not seen, and yet have believed" (John 20:29)

In science, evidence is everything. Ideas that aren't supported by evidence, that don't match up with reality are rejected, however elegant or desirable they are. The ideas that survive give us useful knowledge about the world. In religion, evidence is nothing. Ideas that aren't supported by evidence are proclaimed as truth, and belief in them regardless of evidence is deemed a virtue. The ideas that survive give us nothing of any use.

Religion and Science couldn't be more opposed.