Obviously the treatment didn’t work, but I’m just absolutely puzzled as to how they came up with the idea of it in the first place: Because let’s be serious, how do you go from “incessant diarrhea” to “clearly we need to wash it out with paint thinner”?
Turpentine enemas were, in fact, used to treat patients suffering from what was then referred to as either the Asiatic cholera or the blue cholera, though contextually they do not appear to have been common. Now, problem #1 -- the nineteenth century was a big century, especially for medical care, and treatment and understanding of cholera was different during the second cholera epidemic (when cholera first swept across Western Europe and crossed the Channel to the British Isles) and the third (the last major cholera epidemic in Europe; think John Snow and the Broad Street pump). Let's talk about the second epidemic, roughly 1829 - 1837, which is probably the period Sawbones is talking about.
I'll first refer you to this book, with one Joseph Wallace of Belfast in 1836, who appears to have been a homeopathic physician (https://collections.nlm.nih.gov/ext/cholera/PDF/101223439.pdf). On page 12, he quotes a letter from a Seaton Reid that lays out a treatment strategy, with comments added by Wallace:
When the collapse comes on rapidly, and the heat of the body, despite of the bottles of water applied to it,[there is no reference made here to the immense quantities of poisonous rubbish poured into the or that they should be expected to do any good or bring back patient's health] declines fast, one or two quarts of warm gruel, with a couple of drachms of turpentine, are to be thrown into the bowels by the enema apparatus till the abdomen appears distended, the object of this large injection which may be repeated once or twice is to restore warmth to the system and give a brief support to the exhausted body [and Pharmacopoeia he should have said.]
Reading on, the rationale for turpentine in particular appears to be that cholera depletes the heat of the body, and makes it difficult to bloodlet the patient (the standard of care). By using a turpentine enema sparingly, the patient's internal "heat" will be restored, and blood can be removed.
And in fact, a fascinating case study in the Lancet by a physician named Edward Joseph in 1836 (https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(02)86134-7/fulltext -- you need institutional access) showed just this: John Chapman, an otherwise healthy 21 year-old man fell ill with cholera. Joseph was unable to bleed him, so he rubbed turpentine on his body, and then gave him an enema. With this, he slowly improved, made urine, and was able to be bloodlet. Joseph knew it was working when warmth returned to his body.
Did these therapies work? Absolutely not. I think there's little question the contemporaneous treatments for cholera likely killed more people than they helped, and Wallace gleefully points this out in the section of homeopathic cures. Interestingly, O'Shaughnessy and Latta actually did invent intravenous fluids during this period, though it was rapidly abandoned and morphine/opium, calomel, and bloodletting remained standards (I did a podcast on that, since you seem to be the podcast-enjoying type: http://bedside-rounds.org/episode-25-salt-water/).
But I wouldn't characterize their treatment as patently absurd. This period is one of the most fascinating epistemic changes in the period of medicine; rather than see this treatment as absurd and horrific, I prefer to look it as mostly good-intentioned (though often arrogant) practitioners struggling with the traditional vestiges of humoral theory, new attempts at nosology, the beginnings of using physiological methods, and occasionally stumbling into effective therapies (turpentine, however, was not one of them). I find this is a much more satisfying way to approach medical history.
A finally, I will go ahead and promote my podcast, which I make in partnership with the American College of Physicians. It's largely about this subject -- how modern medicine was made, and the implications for medical practice today. I have a new episode coming out on Monday about the sanatorium movement and the conquest of tuberculosis long before antibiotic therapy existed. It's called Bedside Rounds (http://bedside-rounds.org/), and you can even get CME/MOC for listening (https://www.acponline.org/cme-moc/cme/podcasts/bedside-rounds).