What you are experiencing is a pretty common thing for someone who only rides lesson string horses. It is frequently not so much that the horses can not physically do more than 2'-3" but more that because they are lesson horses there is no need for them to do over 2'-3". Any lesson barn I know of basically says if you want to get past that level you have to buy your own horse.
Lesson horses work hard for a living and their legs and bodies do not need the extra wear doing bigger jumps can cause.
Before you switch barns make sure any barn you are going to will allow you to jump higher on a lesson horse.