When I was in high school, I co-oped on a farm who had a mare who had broken her leg when she was 10. She had been an eventer and was going prelim. Well she hurt herself in the field one day, it was a clean break so they slowly nursed her back to health. She doesnt jump any more, but she does 3rd level dressage and wins consistently.
So yes, a broken leg can be fixed and the horse can have a full (or close to it) recovery.. but I think it also depends on where the break is and what it is (multiple or just one clean break)