It is not possible to quantify this or to give a satisfactory answer. After all, how would we answer the question, how often do bar fights occur in the West today? Where - Tacoma? Phoenix? San Francisco? Denver? Each place has its own profile, and even then, there is a world of difference between a posh hotel bar, a singles pick-up bar, a biker bar, etc. Whatever generalization we arrived at would create the wrong impression and not apply universally - within even one community let alone a region so diverse and large.
That said, there was clearly a lot of romanticism about bar fights in Wild West saloons, and this was not something invented for the twentieth-century film and television industry (where "Westerns" seemed to always have an obligatory bar fight). Here is a portrait of a bar fight dating to 1863: travel writer and artist, J. Ross Browne gave us his impression of the "Cozy Home" saloon, with the charming caption, "Home for the boys." This appeared in Harper's Weekly, a national magazine, but he was clearly pitching his portrait of Virginia City in its earliest days to a nation that already had a stereotypical image in mind when it came to the Wild West and its violence. Some of this was grounded in the fact that the West - particularly the Intermountain Mining West - was predominantly young, male, and single, and when a lot of single young men are drinking in a confined place, violence can result. Still, very few people go to a bar with the hope of getting in a fight, having their face smashed in - or better yet - hoping to be stabbed or shot. Most of the time, the Wild West was a quiet place, but violence is always celebrated more than peace in literature and in any form of media.