Horses were here before the Spaniards:

“There is a documented report from a French baron fighting the Iroquois in 1687 near Rochester, New York, that reads, “In all these villages, we found plenty of horses, black cattle, fowl and hogs.” So much for the fourth-grade idea that Eastern Woodland Indians always walked and that the first horses came to the Plains tribes, who acquired them in the West in the late 1700’s or early 1800’s” (Judith Dutson, Storey’s Illustrated Guide to 96 Horse Breeds of North America, 2005, p. 18).

Horses lived contemporaneously with the American elephant (Mastodon) and the ground sloth (Cureloms & Cumoms):

“The greater portion both of the entire skeletons or extinct animals, and the separate bones, have been taken up from black mud, about twelve feet below the level of the creek. It is supposed that the bones of mastodons found here could not have belonged to less than one hundred distinct individuals, those of the fossil elephant (E. primigenius), to twenty, besides which, a few bones of a stag, horse, megalonyx, and bison, are stated to have been obtained” (Charles Lyell, Travels in North America in the Years 1841-1842, New York: Charles E. Merrill, 1909, pp.139-144).

Deconstructing a Eurocentric Myth

“This research project seeks to deconstruct the history of the horse in the Americas and its relationship with the Indigenous Peoples of these same lands. Although Western academia admits that the horse originated in the Americas, it claims that the horse became extinct in these continents during the Last Glacial Maximum (between roughly 13,000 and 11,000 years ago). This version of “history” credits Spanish conquistadors and other early European explorers with reintroducing the horse to the Americas and to her Indigenous Peoples. However, many Native Nations state that ‘they always had the horse’ and that they had well established horse cultures long before the arrival of the Spanish” (Yvette Running Horse Collin, “The relationship between the indigenous peoples of the Americas and the horse: deconstructing a Eurocentric myth,” Dissertation (Ph.D.), University of Alaska Fairbanks, 2017).

See Elephants and Research for additional references.