They were extinct, that's a well known fact.
Here are some quotes taken from an article called
Horses in the Book of Mormon:
"Horses are mentioned only once in the land northward during the Jaredite period—that is, during the prosperous reign of King Emer around 2500 B.C. and before the great drought sometime in the third millennium B.C. (see Ether 9:19, 30—35). Since horses are not mentioned again in the Jaredite record, it is possible that they became extinct in the region north of the narrow neck of land following that time.
"Horses were known to some Nephites and Lamanites from about 600 B.C. to the time of the Savior. They were found in the "land of first inheritance" during the time of Nephi, son of Lehi (see 1 Nephi 18:25), and in the land of Nephi during the days of Enos (see Enos 1:21). They were also utilized by at least some of the Lamanite elite during the days of King Lamoni in the same general region during the first century B.C. (see Alma 18:9—12). The text does not mention horses in the land of Nephi after that time. The only other region associated with horses was the general land of Zarahemla at the time of the war with the Gadianton robbers, just prior to the birth of Jesus Christ (see 3 Nephi 3:22; 4:4; 6:1). There is no indication in the text that horses were indigenous to that region. The Savior's reference to horses in 3 Nephi 21:14 is a prophecy of the latter days and need not be interpreted as referring to Nephite horses.
In the Book of Mormon, horses are never mentioned after the time of Christ.
"In short, the Book of Mormon claims only that horses were known to some New World peoples before the time of Christ in certain limited regions of the New World. Thus we need not conclude from the text that horses were universally known in the Americas throughout pre-Columbian history. . . ."
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"Even if horses had been abundantly used and had been a vital element in the culture of Book of Mormon people (a claim never made by Book of Mormon writers), one cannot assume that evidence for this would be plentiful or obvious from the current archaeological record. . . ."
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"The horse was the basis of the wealth and military power of the Huns of central Asia (fourth and fifth centuries A.D.). Nonetheless, according to S. Bokonyi, a leading authority on the zoological record for central Asia, "We know very little of the Huns' horses. It is interesting that not a single usable horse bone has been found in the territory of the whole empire of the Huns. This is all the more deplorable as contemporary sources mention these horses with high appreciation."5
"The lack of archaeological evidence for the Hunnic horse is rather significant in terms of references to horses in the Book of Mormon. During the two centuries of their dominance, the Huns must have possessed hundreds of thousands of horses. If Hunnic horse bones are so rare, notwithstanding the abundance of horses during the Hunnic empire, how can we expect abundant archaeological evidence for pre-Columbian horses in the New World, especially given the scant and comparatively conservative references to horses by Book of Mormon writers? . . ."
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"Given the limitations of zoo-archaeology, and also those of other potentially helpful disciplines when probing many centuries into the forgotten past, it is unwise to dismiss the references in the Book of Mormon to horses as erroneous."
