The only time I was on a horse was a pony ride when I was very young. There were horses going in a fenced circle with other children and I was scared and crying for my mother to get me off. The horse behind me was coming up alongside and his head was right next to my leg. Since then I have petted a couple, and although I think they're beautiful animals, I never was really around them and still fear them to a degree.
I've brought my dogs to a large natural park for decades now, and I've seen a few horses become spooked around the dogs, throw off their riders, and run fast and frightened through the park, I had to pull my dog and get out of one's path or be trampled. Now, the horses are not allowed to come in the dog area, and I think it's safer for the horses, their riders and the dogs and their owners. We still get to see them often, just not too close.