We could debate the reasoning that led the Western and Northern Europeans to militarily support the Byzantines until the heat death of the universe - but it's not a particularly interesting discussion.

But the Crusades did spark a lot of in-group / out-group violence in Europe itself. De-tangling the Crusaders related pogroms from the base rate of pogroms in Europe is very difficult - but it is at least plausible that the increased religious fervor was a partial cause of the Crusader pogroms.