Yes indeed, they do. I don't think there is anything wrong with it as long as you consider that they are more constricting than english saddles and your horse is not able to bend and move in them as much and most of them weigh much more than english saddles. Small jumps every now and then are fine, I believe but you would have trouble trying to do a 3 foot jumper course in one.
Please don't, just don't, my friends horse did his back out by doing jsut a little jump in one, maybey the smaller lighter synthetic ones but please go bareback or buy a cheap fitting secondhand english
I heard a horror story once that some girl was jumping in a Western Saddle and somehow got the saddlehorn through her chest and stuck under her ribcage when she went into jumping position over the jump.. When the horse came down on the other side it ripped her ribcage out of her chest. ._.
I've only had a western saddle on my QH once for a trail ride. A tree was down on the trail and we all had to go over it - no problem b/c we event - it's was about 2'6"ish, pretty wide and we made it over fine. I just had one hand on the reins and put my other hand over the horn so it didn't bump against me too hard.
I would not recommend it though, but it can definitely be done.
There are people who do it, just as there are people who do anything, but I wouldn't recommend anything over a foot or two.
I've seen some nasty things happen due to the horn and the swell on that type saddle. I was jumping a rather large log some years ago and got out of position. When we completed the jump, my belt got caught on the saddle horn. The wreck that it caused wasn't pretty.