I find the claim that it is a religion of peace is absurd as well if you read translations of the Quran it has been translated to tell you many cases in which you should kill a person. the same is true for Christianity, in many places if you read their bible it give exception to the thou shalt not kill I think the most famous exceptions come from Leviticus. Both religions depending on how strict the people are who follow them can be considered the instigators of mass violence and death.

I think what happens often is you get peaceful people who think they are a good representation of what their religion is and who don't really grasp what it is to be either Christian or Muslim these people usually often have their own renditions of what God or Allah is to them as well religion really lets these people go around thinking their god is a god of peace and love and they tend to when times get tough lay out their reasons for being violent by making it look to followers like self defense or protection of their collective version of God which nobody can seem to agree on what is even when the gods they worship stem from the same idea.

Religion leads to differences of opinion, differences of opinion combined with faith lead to a deadly chem cocktail just waiting to be ignited called religious contrast once ignited it becomes religious conflict. and if it goes long enough before ignition it can become full out religious war. Ironically war which would be against the views and teachings of any religion that is ever allowed get this far as war is filled with sins against what these religions would call god.