On the wiki it says:
It is possible to jump over wooden fences while riding a horse.
However, will a wild horse also jump over a fence (1.5 blocks)? If so, what can I use instead to contain them?
No, they won't. That jump can only be controlled by you once the horse is tamed by pressing the space button while riding it. But if you want to be 100% sure a horse won't leave any place you can still use a lead and tie the horse to a fence.