Author's Saving Throw: John Green skips over the Protestant Reformation in World History and the Space Race in US History. The closest we get to the latter? An image in the introduction briefly mentioning the Apollo Program and a still image of an Astronaut in the episode on America in the 1960s. In part to respond to this criticism, they have since announced Crash Course World History 2.
Hank manages to pull this off with one to Aristotle in Philosophy, with many of the YouTube comments thanking him for doing this as John would rather shut down Aristotle like he usually would.
Base-Breaking Character: John can easily be seen as this, with the majority of the tropes on this page focusing on his errors and numerous take thats, which some find more insulting than funny. His laser-like focus on social history can either be a unique perspective not shown in normal history classes, a worthy addition but given too much focus, irritating because he tends to say the same thing over and over and, or end up being really insulting.
The episode "Battles of the Civil War" has proven to be this. Some viewers accepted John's basic listing of major battles of the Civil War as a good Take That, Audience! towards those who ask for more military history on the show and acceptable filler to pad out his paternity leave, especially since the following two episodes were also about the Civil War and contained much more analysis and details. Others dislike the episode, either because they see it as disrespectfully trivializing the entire field of military history and/or because an intentionally boring episode is still boring.
The 2016 series on Games is spawning a new broken base for its chronological order and prioritization of topics. First, the decision to skip directly from ancient board games to the advent of video games was met with some criticism, because it ignores a lot of the evolution of board gaming, card games, tabletop games like D&D, and how these all connect to each other. Then, to further break the fanbase, the series focused heavily on consoles over PCs, causing a number of fan debates over what is important in terms of video game history. While the series are now focusing more on those subjects, the fact that the subjects aren't going to be handled in chronological order is a little jarring to quite a few viewers.
In the US History series, John repeatedly claims that Nathaniel Hawthorne was the only author of note in Antebellum America, completely ignoring such writers as Edgar Allan Poe, Herman Melville, or even Washington Irving, not to mention all of the transcendentalist writers that he actually brings up in those same episodes just to mock. This is particularly strange, because not only is John a writer himself, but he also hosts Crash Course Literature.
After two World History series, the show finally covered the Protestant Reformation, but unfortunately, the episode got quite a few facts wrong. Luther didn't come up with the first vernacular Bible, and Green totally skipped over the fact that Luther consciously edited the Bible's wording to fit his own ideas, such as St. Paul saying "faith alone", while the original Greek and later Latin says "faith." No matter what your views on religion, this is a pretty big deal to history. He could have also worked in historical perspective and "who's writing history" into the Reformation quite easily. Was the Reformation a break from a corrupt church? Was it an uprising of violent and intolerant fanatics? Was it a land grab by monarchs whose greed outweighed their faith?
His outright dismissal of Renaissance 'because the lower classes didn't experience it' and his claiming that it was all because of the Arabs has also sparked controversy, especially because the Arab preservation of Greek and Roman works was less important than what many used to think. Aristotle was known well before the Middle Ages were done, and Greeks fleeing the teetering Byzantine Empire brought far more works to the West than translating Arabic texts ever did.
John also didn't do very much in regards to Japanese history in the first series, almost completely leaving out Japan until the invasion of Manchuria, ignoring a large swath of history, such as the Russo-Japanese War and the gaining of the German colonies (an episode on the Heian period came out in the second series). However, he did manage to dedicate an entire episode to the Haitian Revolution, which didn't impact much outside of the island nation (and certainly wasn't on equal footing with Napoleon).
In the "Big History" section, Hank mentions that bonobos are peaceful and "like hippies" because they are female led when compared to the violent male-led chimpanzees. However, the studies Hank is referencing dealt with bonobos in captivity. Bonobos in the wild are just as violent as chimpanzees.
Everything in the Human Geography videos, so much so that they had to pull them from the site and are (as of this writing) redoing them. Said videos denied that the environment had anything significant to do with the development of human culture, attacked the decidedly anti-racist book "Guns, Germs, and Steel" as "super racist", and went on a diatribe against old thinkers (such as Aristotle, a figure John Green hates) whose ideas are no longer in widespread use as if they were representative of current-day cultural determinism. It wasn't Green himself who hosted the episodes, but he apologized for them on Twitter.
Snark Bait: In certain circles (such as the /his/ board on 4chan), the series has made John Green an absolutely despised figure.
Squick: Episode 4 of Crash Course Psychology shows us what happened to Phineas Gage, a man who had a railroad spike go all the way through his head.
Woobie: John Green would probably disagree, but after she's beaten around for an entire Crash Course: World History, it's totally reasonable to feel a little sorry for John's college girlfriend.
Tropes from the TV Series
Paranoia Fuel: Flipping a car onto its roof and being stuck inside. Even with reinforced cars, protective gear, and paramedics on standby...that's still going to be terrifying.