The main written source regarding the conquest of Mexico is the memoir of Bernal Diaz, written in his old age. Diaz served under Cortes for the entire duration of the invasion. Of all the history books I’ve read, this one is probably my favorite, and I learned the most from it. If you have any interest at all in the conquistador period, I highly recommend it.

There is no direct comparison by Diaz of the relative effectiveness of the native’s bows and slings against the guns and crossbows of the Spanish, but readers may still be able to draw conclusions from his descriptions of the battles. Diaz is always sure to mention when a band of Spaniards is armed with muskets and crossbows, and it seems that, besides the horsemen, they were considered the most important troops.

https://www.gutenberg.org/files/32474/32474-h/32474-h.htm

Before the invasion of Mexico, Diaz was a member of two smaller expeditions to the mainland.

As we were thus marching along, and had arrived in the vicinity of several rocky mountains, the cazique all at once raised his voice, calling aloud to his warriors, who it seemed were lying wait in ambush, to fall upon us and destroy us all. The cazique had no sooner given the signal, than out rushed with terrible fury great numbers of armed warriors, greeting us with such a shower of arrows, that fifteen of our men were immediately wounded. These Indians were clad in a kind of cuirass made of cotton, and armed with lances, shields, bows, and slings; with each a tuft of feathers stuck on his head. As soon as they had let fly their arrows, they rushed forward and attacked us man to man, setting furiously to with their lances, which they held in both hands. When, however, they began to feel the sharp edge of our swords, and saw what destruction our crossbows and matchlocks made among them, they speedily began to give way. Fifteen of their number lay dead on the field.

The conquistadors were engaged in battle almost constantly, not just against the Aztecs but nearly every race they met. Compared to the later battles under the command of Cortes, the performance of the Spanish in these early skirmishes was relatively poor. Note that Diaz says fifteen Spanish were wounded immediately by arrows. Although conquistadors are often depicted in popular culture with steel breastplates and morion helmets, conquistadors would not be so armed until decades later. The initial small band who joined Cortes was also made up of the poorest men of the Caribbean colonies. From Diaz’s descriptions it seems like the Spanish on this first expedition had no armor at all, although under Cortes they would wear quilted cotton armor similar to the natives.

Another battle with the natives. Diaz’s expedition was parked on the beach of Yucatan when they were approached by an army that he estimated outnumbered them thirty to one.

They divided themselves into different bodies, surrounded us on all sides, and commenced pouring forth such showers of arrows, lances, and stones, that more than eighty of our men were wounded at the first onset. They next rushed furiously forward and attacked us man to man: some with their lances, others with their swords and arrows, and all this with such terrible fury that we were compelled also to show them earnest. We dealt many a good thrust and blow amongst them, keeping up at the same time an incessant fire with our muskets and crossbows; for while some loaded others fired. At last, by dint of heavy blows and thrusts we forced them to give way; but they did not retreat further than was necessary, in order that they might still continue to hem us in in all safety; constantly crying out in their language, Al calachoni, al calachoni; which signifies, kill the chief! And sure enough our captain was wounded in no less than twelve different places by their arrows. I myself had three; one of which was in my left side and very dangerous, the arrow having pierced to the very bone. Others of our men were wounded by the enemy’s lances, and two were carried off alive; of whom, one was called Alonzo Bote, the other was an old Portuguese.

Perceiving how closely we were hemmed in on all sides by the enemy, who not only kept constantly pouring in fresh troops but were copiously supplied on the field of battle with meat, drink, and quantities of arrows, we soon concluded that all our courageous fighting would not advance us a step. The whole of us were wounded, many shot through the neck, and more than fifty of our men were killed. In this critical situation we determined to cut our way manfully through the enemy’s ranks and make for the boats, which fortunately lay on the coast near at hand. We therefore firmly closed our ranks and broke through the enemy. At that moment you should have heard the whizzing of their arrows, the horrible yell they set up, and how the Indians provoked each other to the combat, at the same time making desperate thrusts with their lances. But a still more serious misfortune awaited us; for as we made a simultaneous rush to our boats, they soon sunk or capsized, so that we were forced to cling to them as well as we could; and in this manner by swimming we strove to make the best of our way to the small vessel, which was now in all haste coming up to our assistance. Many of our men were even wounded while climbing into the vessel, but more particularly those who clung to its side; for the Indians pursued us in their canoes and attacked us without intermission. With the greatest exertions and help of God we thus got out of the hands of this people.

After we had gained our vessels we found that fifty-seven of our men were missing, besides the two whom the Indians had carried off alive, and five whom we had thrown overboard, who had died in consequence of their wounds and extreme thirst. 

…

None of us had escaped without two, three, or four wounds. Our captain had as many as twelve, and there was only one single soldier who came off whole.

It is a common theme in Diaz’s memoir that the spears, arrows and slingstones are able to inflict many wounds, but kill few. With more than 50 killed, the above battle is among the most disastrous for the Spanish.

Recognizing land charted by Ponce de Leon a decade earlier, Diaz’s band landed in Florida a few days later to take on water, when they were attacked again.

With joyful hearts we then took our fill of the refreshing beverage, and washed the bandages of our wounded. A good hour’s time was spent in this, and as we were on the point of re-embarking with the casks of water, quite overjoyed at our success, one of the men whom we had placed sentinel on the coast came running towards us in all haste, crying aloud, “To arms! to arms! numbers of Indians are approaching, both by land and sea.” And indeed the Indians came up to us almost at the same time with the sentinel.

They had immense sized bows with sharp arrows, lances, and spears—among these some were shaped like swords—while their large powerful bodies were covered with skins of wild beasts. They made straightways to us, let fly their arrows, and wounded six of our men at the first onset. I was also slightly wounded in my right arm. We, however, received our enemies with such well-directed blows and musket-shots that they very soon quitted us who had been digging the wells, and turned towards the creek to assist their companions who in their canoes were attacking those left behind in the boat. The latter had been forced to fight man to man, and had already lost the boat, which the Indians were towing off behind their canoes. Four of the sailors had been wounded, and the pilot, Alaminos, himself severely so in the throat. We, however, courageously faced our enemy, went up to our middles in the water, and soon compelled them, by dint of our swords, to jump out of the boat again. Twenty-two of the enemy lay dead on the shore; three others, who were slightly wounded, we took on board with us, but they died soon after.

Arriving back in Cuba, the captain who had taken so many wounds finally died of them several weeks later. The governor of Cuba ordered a second expedition. Diaz’s band returned to the beach where they had lost over fifty men:

They had arranged themselves along the sea shore in order to fall upon us as soon as we landed. But, as our previous loss had taught us prudence, we took with us this time some falconets [light artillery], and otherwise well armed ourselves with crossbows and matchlocks.

When we were near enough they let fly such a shower of arrows and lances that the half of our men were speedily wounded. As soon, however, as we got on shore, we quickly gave them an evil return with our matchlocks and sabres. Nothing daunted by this they each selected their man, whom they particularly aimed at with their arrows, but we had taken the precaution to put on cotton cuirasses. They continued to combat with us for some time, until the arrival of another of our long boats, when we drove them back to the wells near the village. In this conflict we lost Juan de Quitera and many other soldiers. Our commander, Juan de Grijalva, got three arrow wounds and lost two of his teeth, and above sixty of our men were wounded. Immediately upon our putting the enemy to flight, we entered the village, dressed our wounds, and buried the dead.

It seems Diaz thought the cotton cuirasses were effective, but nonetheless many were wounded. Still, this time the Spanish were the victors. Diaz put the number of native killed and wounded at 200.