There was a book in remember reading when I was younger (in the 90s) about a horse that turns into a unicorn. I think it could have possibly been a series. There is also a little girl that I think becomes a princess when they go to the unicorns world. For some reason I remember her dog coming along with them.
That would be The Unicorns of Balinor series by Mary Stanton. The young girls name is Ari, she runs away from her farm home with her horse Chase and dog Link. They end up passing through a portal to a different world where her horse changes into his rightful form, a unicorn. Adventure ensues.