European and Euro-American sources from the 16th-19th centuries, which here include travel accounts and treatises on how to swim, unanimously agree on several points: (a) they swim badly (b) the people of basically every other culture they encounter swim better (c) there is apparently no reason for the white people to go, Huh, maybe we could learn a thing.
George Catlin, who traveled the Upper Plains in the 1830s, distinguished Euro-American swimming style, at least of trained swimmers, from that of the Mandan (Numakaki or Nueta):
The mode of swimming amongst the Mandans, as well as amongst most of the other tribes, is quite different from that practiced in those parts of the civilized world. The Indian, instead of parting his hands simultaneously under the chin, and making the stroke outward, in a horizontal direction...throws his body alternately upon the left and the right side, raising one arm entirely above the water and reaching as far forward as he can, to dip it...whilst this arm is making a half circle [underwater], the opposite arm is describing a similar arch in the air over his head, to be dipped in the water as far as he can reach before him, with the hand turned under, forming a sort of bucket, to act most effectively as it passes in its turn underneath him.
Letters and Notes on the Manner, Customs, and Condition of the North American Indians
Catlin contrasts the European style of swimming, a rudimentary breaststroke, with what is a decent approximation of modern freestyle (forward crawl). At least, with respect to the arms. The form of kicking is not often described in travel accounts and letters, which is interesting.
A similar distinction is noted by Portuguese and other European slavers and visitors to West Africa. French slave trade agent Jean Barbot in the 17th century described swimmers of the Gold Coast:
The Blacks of Mina [Elmina] out-do all others at the coast in dexterity of swimming, throwing one [arm] after another forward, as if they were paddling, and not extending their arms equally, and striking with them both together, as Europeans do.
Once again, the contrast is made between essentially freestyle and essentially breaststroke (unless the French are doing fly, which...no). Barbot does observe that African nations from further inland are not so practiced at swimming, preferring not to swim (for pleasure or for food and travel) in their rivers. This inclusion is actually kind of a change of pace for Euro-American accounts. With respect to the people of Elmina, he further notes a couple of different manners of swimming besides a single stroke variety to propel themselves forward:
...playing on the surges of the sea, about the shore, some on pieces of timber, others on bundles of rushes, made fast under their stomachs, the better to learn how to swim; others ducking under the water, and continuing there for a considerable time
Barbot associates these rafts with learning how to swim, although the description makes it seem like they are also a prop or toy used by experienced swimmers. The idea of lightweight wood as a beginner's tool might reflect European practice. For his Early British Swimming, Nicholas Orme turned up a wild reference in a London chronicle from the 1420s:
Diverse persons of low estate of the city of London, assembling one day on the wharf...wished and desired they had the person of my lord [the bishop] of Winchester, saying they would have thrown him into the Thames to have taught him to swim without wings.
Orme takes this as a reference to water-wings, better known today as "floaties" (the inflatable things you see kids wearing around their upper arms). Even though, as he points out, there isn't another known reference to "water-wings" as the name for a swimming tool until the 20th century, his conclusion about a device of some sort seems more plausible to me than this being a metaphorical reference akin to flying without wings or some such.
But not all Europeans made the same connection, or rather, some were confronted with evidence so strongly to the contrary that they couldn't plausibly believe instruction the only use worth mentioning. Charles Warren Stoddard, adding a bit of LGBTQ diversity to the Dead White Dudes' Club of this answer, described Native Hawai'ian swimming--or rather, surfing--in 1874:
Kahele, having been offered a surf-board that...was itself as light as cork and as smooth as glass...seized his sea-sled, and dived with it under the first roller which was then about to break above his head, not three feet from him. [does the same with more waves] ...Again Kahele dived and reappeared on the other side of the watery hill...turned suddenly, and, mounting the towering monster, he lay at full length on his fragile raft, using his arms as a bird its pinions...As it rose he climbed to the top of it, and there, in the midst of foam seething like champagne...He leaped to his feet and swam in the air.
Summer Cruising in the South Seas - I hope you'll check out this whole passage
Later on, he mentions swimming that seems more like swimming as such, but doesn't provide any descriptive details.
Our white explorers almost unanimously emphasize two further points that intersect with debates in modern scholarship (mostly on different subject matter) over pre/early modern Euro-American swimming. First, they stress that everyone swims, even children ("These islanders are amphibious. The little bronze babies float like corks before they can walk half the length of a bamboo-mat," writes Stoddard). Second, they stress that "everyone" includes (!!!) women.
The first point, about the extent of swimming ability among white men, has several points to the contrary: a lot of people (including, by the way, Stoddard) explicitly say they can't swim; in the later 19th century, there's a whole panic among naval officers and commentators about the lack of swimming ability even among midshipmen at the U.S. Naval Academy; swimming treatises are not only prescriptive but aimed first exclusively and second mostly at the noble/upper classes.
What seems more likely, from other accounts of sailors and sailing, is that there wasn't much formal swimming instruction, as in the strokes or freediving technique, and no training in swimming endurance. However, sailors at least would be able to tread water and other things to keep from drowning if they went overboard in port or some such.
The question of women swimming has a couple of points against the possibilit as well: (a) they are hardly ever mentioned (b) there's often a distinction made between "bathing" and "swimming," in which men swim while women bathe, or religious leaders are like "women can be in the water only to bathe", (c) in early medieval-16th century sources, there is a strong association of swimming and military practice, and (d) most people swam naked.
In medieval sources, at least, there are a few references to women bathing while wearing some sort of smock, and Orme observes that men in Norse sagas (who are very often strong swimmers--see Beowulf) sometimes strip down to their undergarment but no further to swim. In general, however, swimming for recreation follows along with seafarers realizing their ship is about to capsize--the first thing they do is strip.
There don't seem to be that many references to what people wore to swim in the Euro-American travel accounts. The exception in the ones I looked at for this answer is again Stoddard, who is writing after the invention of swimsuits in the west:
One after another, they came out of the sea like so many mermen and mermaids. They were refreshingly innocent of etiquette--at least, of our translation of it...With uncommon slowness, the mermaids donned more or less of their apparel, a few preferring to carry their robes over their arms; for the air was delicious, and ropes of seaweed are accounted full dress in that delectable latitude.
The reference to mermen and mermaids isn't unique; it probably reflects the ease and skill with which the Euro-Americans see people from other cultures swimming, in comparison to their own. This is not a unanimously positive thing. As I mentioned at the beginning, none of these accounts feature the author saying, "Hey, maybe teach me?" Kevin Dawson, who has published some version of the major article on early modern swimming about eight different times, observes that a lot of white authors in the 17th-mid 19th century make a strong connection between swimming ability and animal/natural traits:
Man cannot swim with the same faculty as many of the inferior animals, which seem to be led by instinct to use the proper action for their preservation, while rational creatures, being aware of their danger, grow fearful or impatient, and begin to struggle.
-introduction, 1840 edition of Benjamin Franklin's letter/treatise on swimming
(Note that Franklin, for his part, advised everyone learn to swim). Dawson thinks that probably Europeans applied this dichotomy to the people of color they encountered. That, though, presents a really interesting contrast to what we see in medieval sources, which is a strong connection between swimming and knightly prowess. It would be fascinating to trace whether, when, how, and to what extent that shift actually occurred.
I know this has ranged further afield than the original 17th-18th century time frame. However, I hope the general similarities in accounts over the time span help justify that extension--as well as, quite frankly, how awesome these sources are.