Forget about everything you have ever been taught regarding dressage position. That will cause your dislodgement. If a horse shies to the right, here is what you must do:

twist your hips to the right slightly
lock up your left hip, knee and ankle joints.
Basically have a straight left leg, from stirrup to hip.
Throw your left leg forward of the girth.

Shying Horses

More riders fall off horses due to the horse shying than from horses bucking. Shying is a description for the horse unexpectedly, suddenly and violently jumping one way or the other and is caused by the horse exhibiting fear of something that it has seen come into it’s surroundings.

Some horses shy only because of this natural instinct and some, undoubtedly purposely manufacture things to shy at.

Certain types of riders can suppress or enhance shying, depending upon their attitude, personality and riding style.

“A bold rider makes a bold horse and a nervous rider makes a nervous horse”

The overwhelming problems due to shying are:

A lack of boldness in the rider causing horse to be nervous
A lack of knowledge of how to ride a shy

The moment the horse signals a rise in it's suspicion and fear level, commence on the inside (to the object of fear) rein. You never want to take the horse nearer to the goat or up to the goat; all you want is for him to go past the goat without shying to the outside. Use the inside rein in a direct percentage of controlling the angle of the horse, in order to nullify any chance of run to the right. If you have to end up side passing by the goat, do, butdo not pull on the outside rein. You can even turn the side pass into a leg yielding training opportunity. If you have pulled on two reins youI would be telling the horse that you are frightened because you have done what all nervous riders do.

As well as this rein control use what ever level of energy that is required to keep the horse going forward and then as you feel the fear level subsiding and you pass the goat, go progressively more outside rein and straighten up, continuing on in an un-interrupted trail ride with the minimum of fuss and on a loose rein.
Riding with a tense rein. Holding a contact on pleasure horses is without doubt, the second biggest cause of tension in horses on trail rides to rider’s vibes. It is the cause of the dancing, prancing horses or the world, the piaffe, passage, and rearing. For a horse to be relaxed on a pleasure ride it must be able to lower its neck and a horse cannot lower its neck if it is on a contact.

So, to fix a shyer, one must win each and every battle and as you do so, each new and frightening item is cancelled out. Cancel them all out and you have the perfect trail horse.
Another major mistake that people make with 'shies' is to make a bigger issue of the frightening . They try to force the horse to go up to it, sniff it, touch it. They bully the horse into going up to the feared object causing the horse to remember this object as something to be feared.
Simply get past whatever it is with the minimum of fuss even if you have to get off (if there is a traffic issue, I recommend getting off and leading the horse past it - always get back on as soon as the horse has calmed down and continue on.
One of the worst things people do is to punish the horse for shying. Punishing a shying horse is only teaching him that not only does the object look scary but something bad will happen to him every time he sees something spooky, therefore his spooks will intensify at every scary object. NEVER PUNISH A HORSE FOR BEING AFRAID OF SOMETHING - he's NOT doing it on purpose!

REIN CONTROL

If you make a horse look into the opening of a bridge and no matter how hard the horse tries to evade that bridge ,you negate the horse's attempts at turning it's head away, whilst generating the horse forward to the point where the horse must go forward, then it just has to go over the bridge. End of story.

Now the rein control for this exercise is definitely two open reins. Your forearms are still in proper dressage position, the rein hands travel past the hip, but your hands should be set a little wider. So, as fast as the horse tries to evade left and right with its head, the rein hand negates the opportunity and cancels out the attempt. Only one rein is used at a time with the other loopy. Never pressure on both side of the mouth. No different to loading onto a float. If the front handler simply does not allow the horse to ever look around the side of the float and the rear handler, say the whip person can generate the horse to go somewhere; the horse must go in the float.

Evading the entrance of a footbridge is however, a continual series of shies, together forming the evasion. Rein control stops evasion and whip, spur etc, provides the forwardness to complete the rider's wishes. Sophistication should be added however, rewarding the horse for slight forward movements, allowing the horse to put it's head down to sniff, etc. Incidentally, this will almost always happen, just before the horse proceeds and is a sure sign that the horse has already decided to cross. Providing you don't make any mistakes at the last moment.

When a horse is facing directly at the bridge opening, you must be giving with both reins. No contact on the mouth at all. This is very important!

How do we ride a shying horse? Why do so many people fall off them?

You must temporarily forget your dressage training, the moment you run into trouble out on the trail or anywhere else for that matter. Dare I say it, even on the Dressage Arena. So to ride a shying horse, you must ride like the rider of a "Cutting Horse". After all, the cutting horse has to be the most violent shying horse in the world and the Cutting Competitors' have to be the best riders of shying horses.

When a horse shies to the right, you are forced off the left side of the horse by the opposite force. The major reason good riders' don't fall off the horse that shies is their ability to read the event before it happens. You are then prepared. The other reason is reaction time. The more you ride them the better you get at it. Finally, the way you use your position and how you hold your legs has a major bearing on your stability.

Forget about everything you have ever been taught regarding dressage position. That will cause your dislodgement. If a horse shies to the right, here is what you must do:

twist your hips to the right slightly
lock up your left hip, knee and ankle joints.
Basically have a straight left leg, from stirrup to hip.
Throw your left leg forward of the girth.

Now, as the force that is throwing you off the left side of the horse commences, you will have locked against it and become part of the horse, which, as it travels to the right, catapults you to the right with it. The left stirrup iron becomes the slingshot and you the missile. Only thing is that you are traveling in the same direction and at the same speed as the horse and arrive simultaneously at the same location.

As I said, the cutting horse is the best continuous shying horse of all and the cutting rider is simply reading the impending change in direction, locking the appropriate leg and being catapulted with the horse. It matters not what the other leg is doing at all.