No great mystery there. In most disciplines, children are not permitted to show stallions. In most states, you have to have special facilities to even keep a stallion on the property.
Like dogs, the majority of male horses are neutered at a young age for the safety of the average handler. Unlike dogs, spaying a mare is a complex and expensive procedure. Therefor, most female horses encountered by the average rider are intact and most male horses are gelded.
I would not say most “young girls” prefer geldings. I would say most people prefer geldings. As intact mares compete for “alpha” status and go into heat several times a year, most recreational riders (men and women) prefer geldings because they are less likely to challenge the authority of the handler. The majority of riding school “lesson horses” are geldings (I would be surprised if you could find a riding school with more than 20% mares in its school string).
As you move out of the beginner ranks you find an increasing preference for mares as the skill of the rider increases – many people think a mare has more “fire” and intelligence in her performance, where a gelding is dependable but dull.