Just about is fair but still short by ~400 years, unless you count the Jaredites. They purportedly fled the Old World around the time of the Tower of Babel (~2000 BCE IIRC?).

EDIT: I wasn't sure if the Book of Ether even mentioned horses, but it did, along with elephants. [1]

[1] https://www.lds.org/scriptures/bofm/ether/9.19?lang=eng&clan...