One has to assume that it was no treat being injured or ill aboard a seafaring vessel of any kind "back in the day". I certainly wouldn't have wanted to try it myself. Physicians and surgeons, though in fact "trained" as early as the Renaissance, were actually educated through an apprenticing system. Even actual physicians who walked the wards at teaching hospitals were not going to school in the sense that we think of it now. They were observing other doctors at work and then - if they were fortunate - putting their observations to work on illegally gotten cadavers.
Most navies, by the 18th century, had a rank of Surgeon. The ratio in the Royal Navy was one surgeon for each ship that carried 70 men with a surgeon's mate (usually a man of equal capability but, perhaps, lesser experience) added for each additional 200 men. An 800 man ship of the line could ship a surgeon and four mates along with assistants, or loblolly boys as they were known. This meant that "physic" was surprisingly available to men at sea, but many times the cure was as bad as - or worse than - the disease.
While surgeons were so called because they "did surgery" (a nice euphemism for "cut stuff off") the reality was that battle just wasn't as frequent as Hollywood would like us to believe and therefore truly gruesome wounds were not as common as one might think. Of course there were accidents. Falls were a concern but, unfortunate, most of the victims of these sorts of mishaps died out-right. There were burns, from galley stoves and gun practice, which were generally treated with olive oil and could become infected. Sprains and strains were common and commonly faked as well for a few days in the hammock. The vast majority of the surgeon's work, however, focused on illness.
By the early 19th century, a surgeon's medicine cabinet (usually looking very similar to the one above) would include treatments for fever, purging, pain relief and STDs.
The first was a tonic made from cinchona which was usually referred to as Peruvian bark. The bark was brewed into a brown tea and it contained quinine which, as we now know, does in fact help alleviate malaria symptoms. Because it worked on the fever of malaria, surgeons assumed it would work on any other fever, too. In those cases, however, it was simply a placebo.
Purges could be administered to induce elimination or vomiting, depending on the ailment. Enemas were common. Medicinal rhubarb, Castor oil, tartar emetic and ipecac were all used by mouth. Sailors were notoriously fond of purges for some reason, and surgeons kept these goodies under lock and key.
Pain relief was achieved via alcohol originally and then opium tinctured in alcohol, which is known as laudanum. Both were known to be addictive, but considering the potential pain of having an arm sawed off, no one seemed to give that much thought.
Finally STDs, which at the time were strictly considered gonorrhea and syphilis, were treated with mercury. There were oral preparations in the form of daily pills and mercuric chloride suspended in ointment for topical application. Intractable cases were treated to mercury salts injected directly into the ureter. All of these were used on both men and women, although syphilis in particular was hard to diagnose in females.
Then there were the truly grim treatments, pulled from the handy medicine chest only in life threatening situations. Blistering of the skin through the application of plasters made from mustard or cantharides beetles were used for high fevers, the thought being that the "heat" would be drawn to the painful blister formed by the plaster. Bleeding was also frequent. A surgeon would take approximately 12 ounces at a time, but much more could be collected from someone suffering from dangerous diseases such as typhus or yellow fever. Those who survived the disease and the treatment were the strongest of the strong.
Finally, there were times when this treasure chest had to be opened:Saws, knives, scalpels, syringes, probes, retractors, along with needles and cat-gut for sewing things up were all at hand for when the much feared fester of gangrene set in or when a crush injury would surely lead to infection. Of course anesthesia was not an option and most surgeons preferred to offer pain relief only after the procedure was done, for fear of trauma to the patient (I am not making that up). A man's mates or the surgeon's assistants would hold him down. A good surgeon could remove a lower leg in four minutes, bone and all.
While well funded privateers might ship surgeons routinely - Sir Francis Drake and John Paul Jones are good examples - most pirates got by with either a pilfered or make shift medicine chest. If a prize with a surgeon aboard was taken, Bones was usually "encouraged" to volunteer aboard the pirate's vessel. A notable exception is Alexander Exquemelin who voluntarily shipped with both Francois L'Ollonais and Henry Morgan as surgeon. Failing any of that hey, the carpenter had saws.
The history of medicine is frequently bleak, and I've no doubt that the seafaring surgeon's work was often grim and bloody. One thing is for sure, though. Surgeons were almost universally venerated by seaman of all ranks and levels of education, as if they were ancient shamans or medicine men. I guess, to some degree, they were. But no bleeding for me, thanks.