A turtle tank and a fish tank have the same chemistry and a very different load. Everything on the aquarium pages applies here, and the reason turtle water goes wrong faster is not mysterious: a turtle eats more, produces more waste, and often eats in the same water it lives in.
The nitrogen cycle is a property of the water and the filter, not of the animal in it. Waste releases ammonia, one group of bacteria turns it into nitrite, another turns that into nitrate, and you remove the nitrate by changing water. Merck's veterinary reference covers this as environmental disease of aquatic animals generally, not of fish specifically.
Ammonia behaves the same way too. The proportion that is in the toxic form still depends on pH and temperature, so a reading from a turtle tank means what it means for the same reasons, and you can put it through the ammonia calculator exactly as you would for fish.
Three things, and they compound.
Almost all the water quality research is about fish, because it was funded by fish farming. We could not find published ammonia, nitrite or nitrate thresholds for aquatic turtles from an extension service, a veterinary reference or a regulator. The chemistry carries across. The thresholds are fish thresholds, and anyone quoting you a precise safe nitrate figure for a red-eared slider is quoting something nobody measured.
The species numbers that are published, including water and basking temperatures, are on the species pages.
Red-eared sliders — the commonest pet turtle, and the one
most often bought at a size that gives no clue how big it gets.
Axolotls — fully aquatic, and cold water animals, so the usual
tropical advice about heaters is the wrong advice for them.
Bettas — the fish most often sold for a tank with no filter at all.
In England and Wales the Animal Welfare Act 2006 covers "a vertebrate other than man", which includes turtles, and requires that an animal's needs are met including "its need for a suitable environment". The government's guidance for anyone selling pets requires that water quality "must be tested and recorded ... at least weekly". Nothing in either document is limited to fish.