Join Date: Aug 2009
Location: New South Wales, Australia
Under 4ft isn't really considered that big. I'm on the opinion that most athletic horses, provided they are correctly conformed and trained, are capable of jumping that height. I knew a little 14.1hh pony who could easily clear those heights, and was a very competitive eventer in the local area.
Many Thoroughbreds tend to excel at those heights, provided they are sound, because it does not require a great amount of jumping skill, and their speed can be a real benefit. I personally do not like TBs though, I think you'd be better of with a QH
Quarter horses may not be bred for jumping, but I think many of them can clear those heights. A while ago Australia entered a QH/TB cross in the Olympic Showjumping, I believe, and years ago the world champion for six bar was a QH. Their ability to turn may even give you an advantage on some courses.
If you are serious about showjumping, at high levels, you would look for a purpose bred horse, usually some sort of warmblood, who has been intentionally designed for that purpose. They are usually pricey - but you're going to need money to get to that level anyway.
Otherwise, there are many riding breeds for that height range.