These are not very well supported by first-hand accounts, for example in my favourite history book of all times, The Conquest of Mexico, by Bernal Diaz del Castillo (one of Hernan Cortes' men), there are many descriptions of battles between the Conquistadores and the Aztecs and the Aztecs don't run away - the gunpowder and the horses make a great impression on them but they stay their ground and fight. Except, in Bernal Diaz's telling anyway, they are used to fighting ritualistic battles, where the point was to capture some of the enemy's men, to later offer up as sacrifices to their gods. So the Aztecs are playing for points, in a sense, whereas the Spaniards are fighting dirty and going for the throat. Obviously, the Spaniards have the tactical advatage and it's not because of their weapons and armour, or their steeds.

Another detail almost always overlooked is that the Spaniards were allied to local Mexicans that were enemy to the Aztecs of Tenochtitlan, particularly the people of Tlascalla. So in many of the battles with the Aztecs, the few hundred Conquistadores Cortes has with him are joined by thousands of Tlascallan warriors. This information is almost always left out of the narrative of the overwhelming technological superiority of the Conquistadores that allowed them to win despite being fewer in number. In truth, the forces of the Conquistadores joined by the Tlascallans and other enemies of the Aztecs were comparable in size with those of the Aztecs.

Finally, when the Aztecs decide enough is enough and start fighting for real, the Conquistadores run for their lives. This is the tale of La Noche Triste, The Sad Night, when the Conquistadores try to escape Tenochtitlan under cover of darkness with all the gold they've looted from Moctezuma's vaults, after he was killed (by his own subjects, according to Bernal Diaz, but, well... he would say so, wouldn't he?). They are chased through the streets of the city by the Aztecs who are in open revolt and catch and kill them one-by-one. Most drown in the waters around the city, many weighed down by the loot they cling to, stubbornly. Just a few barely escape with their skins and some of the gold. It's a sad night indeed- and it really puts the kibosh on the Conquistadores' crushing superiority, technological or other.