I only count 13, 2 of which are in Isaiah, and a 3rd one where Jesus is either quoting a prophecy or making a prophecy. So there are 10 valid references to horses in the New World.
"And they said unto him: Behold, he is feeding thy horses. Now the king had commanded his servants, previous to the time of the watering of their flocks, that they should prepare his horses and chariots, and conduct him forth to the land of Nephi;" Alma 18:9, about 90 BC.
So the king has horses and chariots, both of which he uses to travel across the country. I believe the horses were used as pack animals, and the chariots were litters, carried by men.
Now when Lamoni had heard this he caused that his servants should make ready his horses and his chariots. And he said unto Ammon: Come, I will go with thee down to the land of Middoni, and there I will plead with the king that he will cast thy brethren out of prison. - Alma 20, about 90 BC
Once again, it is the king who owns the horses, which he uses to travel as part of an entourage.
"And they also had horses, and asses, and there were elephants and cureloms and cumoms; all of which were useful unto man, and more especially the elephants and cureloms and cumoms." - Ether 9:19, aprox. 2600 BC
Apparently, the horses were only marginally useful.
"And now it came to pass that the people of the Nephites did all return to their own lands in the twenty and sixth year, every man, with his family, his flocks and his herds, his horses and his cattle, and all things whatsoever did belong unto them." - 3rd Nephi 6:1, aprox. 26 AD
So, some people owned horses, but there is no indication how many or what they were used for.
"And it came to pass that the people of Nephi did till the land, and raise all manner of grain, and of fruit, and flocks of herds, and flocks of all manner of cattle of every kind, and goats, and wild goats, and also many horses." - Enos 1:21, aprox. 420 BC.
The implication is just the opposite; horses were generally scarce.
The other references are to these same four events, and don't shed any further light on the number of horses or what they were used for.
We may be talking of hundreds of horses, but not thousands. Horses are never mentioned pulling wagons, or used in war. No one is ever mentioned riding a horse. I think they were used as pack animals.