Horses definitively existed in (far enough) precolumbian America. The dominant belief seems to be that they died out (perhaps due to human hunting) at or shortly after the end of the Pleistocene, ca. 10-12kya, but there is some evidence there may have been some populations around as late as ca. 3kya, IIRC.
EDIT: of course, one must keep in mind that the horses of precolumbian America were much smaller than Old World horses, so it's not really plausible that they'd be being ridden, pulling chariots, or generally being used like Old World horses even if it is (just barely) plausible that they were around at the right time.
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...
And was that peer reviewed?
"How a Mormon lawyer transformed archaeology in Mexico—and ended up losing his faith"
http://www.sciencemag.org/news/2018/01/how-mormon-lawyer-tra...
This has made it more difficult for the church to say "look, we were right!" when discoveries like this are made because the timeline in several cases doesn't align with 600 BC to AD 400.
https://en.wikipedia.org/wiki/Archaeology_and_the_Book_of_Mo...