Maybe something like that would help you conquer your fears. I know that a long time ago when I worked at an animal shelter for a few months, I was afraid of specific dogs, like pit bulls, because of all the news media stuff plus my own personal bad frightening experience ONE time with one.
But then because I had to work with them everyday, I began to see that the pits that were available for adoption were beautiful wonderful friendly sweethearts. One was just so friendly that she liked to be held like a baby with her feet up in the air, and she wasn't small! Doesn't mean that I wasn't normally cautious and looked for signs of aggression, but by exposing myself to them helped me conquer my fear. At first, I was literally shaking in my shoes, and I was so afraid and every hair on my head stood up, and sometimes I thought I was going to literally freak. But I was also afraid of showing that fear in front of the dog for fear that my own fear would cause them to attack. So I never acted afraid, and I began to realize I didn't have any reason to be in the first place.
I was taught how to handle things and it helped me so much, and I found a new appreciation for not only pits, but for big scary type dogs in general.
I remember in the beginning of the interview when I first applied, they asked if there was a particular animal i was afraid of. I just said, yes...big scary dogs! They laughed and said, they understood, and that I would be able to have someone help me at first deal with them. So though I wouldn't go into an interview saying you were scared of horses, you may rephrase it to say you look forward to gaining more confidence in dealing with horses.
I would spend any extra free time I had during the workday picking out dogs I was scared of and walking them - at first I would have someone else come with me, so maybe you can do the same sort of thing - at first just try to hang out with the people that hang with the horses to learn more and feel safe.
That way you can prevent a great job from slipping away.