Training your horse to cross over obstacles is best done in a controlled environment with the least number of possible distractions. Horses are smart animals they can learn pretty quickly. Therein also lies a particular drawback that can make training them more than challenging: smart animals have attitudes.
Horses would not be able to perform complex and graceful Olympic equestrian manoeuvres if they weren’t smart. But ask anyone who’s trained a horse for dressage and did so successfully and they’d tell you one of the first hurdles they had to overcome was either subdue the proud animal’s attitude or meet it halfway, the latter more advisable than the former.
Your horse wants to be left alone whilst doing what is required of her, although you unfortunately can’t just let her do that, unless you’ve already established through a cue exactly what it is that you want her to do and how. You can teach her to place her feet where you want them, either whilst from the saddle or from the ground, but whatever cue you want her to be familiar with, just remember to release her from it quickly after she responds correctly.
Regardless of the obstacle, you need to be decisive on exactly where you want her to cross. Being more specific makes it easier for her to understand. The key point to remember is to always keep her eyes on that focus spot. Try applying your “go forward” cue and release her from it as soon as she responds.
If you’re focusing on the “go forward” cue, you will need to counteract all other motions she tries while keeping both her eyes on the same spot as before. Move her back to the left if she moves to the right and vice versa. She’ll will keep moving her head away from the point of focus—just keep turning it back to the spot where you want her to cross. When you notice her pawing or lowering her head, she’s thinking about going forward, so apply your cue and reward her responsiveness if she does go forward.
When she reaches the obstacle, allow her curiosity to be satisfied by inspecting it. If the obstacle is water, just let her paw it; if it’s step ups, let her go a foot at a time; for crossovers, let her get a feel for it. The first “step” is always the biggest, and if she successfully clears the obstacle then start over from the other side. Repeat until she crosses without hesitation to the other side and back. It’s advisable to work using an obstacle your horse would think she couldn’t cross in an area sufficiently large to allow for safe manoeuvring.