Now, I know I can't speak for all transgender individuals, but I doubt that transgender children "grow out" of their transgender feelings. There are some children who cross-dress, which is a completely different thing from transgenderism. If a child does "grow out" of their transgender feelings, then they're likely to look back on why they had transgender feelings in their teenage or adult life, and see where that takes them.
In my experience, I had transgender feelings at quite a young age, particularly my 3rd and 4th grade years. I mostly forgot about them through my middle school years, but still felt discontented with myself. It was during my late high school years when I was able to piece my feelings together.
Something that radical Christian fundamentalists like to do is claim that the transgender community is hurting children (whilst supporting the NRA), and take the moral high ground. They have a hard time accepting new scientific discoveries, and fill in the blanks with whatever they can find from the Bible. Whenever new sciences are discovered, radicals will do whatever they can to keep themselves in power, specifically use the money they've received from donations (supposedly to help the needy) and become lobbyists for the more conservative politicians, promising to use their money to promote the politician, while the politician puts forth conversion therapy.
Yeesh... I went on a bit of a rant there. Thanks for asking! :)