While wandering the Earth Kingdom in the beginning of season 2, Iroh, the Eccentric Mentor of Avatar: The Last Airbender, comes across a plant that is either "the rare White Dragon Bush, whose leaves make a tea so delicious it's heartbreaking!", or else "the White Jade Bush, which is poisonous." You'd Expect: He'd demonstrate the good sense possessed by even the most scatterbrained Cub Scout and not touch the thing with a ten-foot-pole until he's 100% certain it's the right one. Instead:He drinks it and almost dies. Added Stupidity: He later finds a plant that is either the antidote, or a plant that can cause blindness. Considering that he was DYING, you'd think that being blind isn't that bad. After all, you don't need eyesight when you are dead, so it's essentially a 50% chance of healing with (basically) no downside. He doesn't try it though. Granted, it was Zuko who stopped him from doing so, but Iroh still took the time to THINK about whether or not he wanted to, as if it weren't incredibly obvious. Not to mention that he didn't try reasoning with Zuko at all.
During one of their trips through the Earth Kingdom, we had a crazy general trying to force Aang into the Avatar State. You'd Expect: The General would NOT try to piss off the AVATAR. Or to just realize that you CANNOT control a supernatural force. Instead: He has his men attack Aang for twenty minutes of the episode, and when he finally does go into the Avatar State, the crazy bastard doesn't seem to mind that he and his men are getting their ASSES WHOOPED. Even when Aang's finally finished his mad rampage, he wonders out loud how to control him when he's in that state. Thank you for interfering, Sokka.
Long Feng, the Evil Chancellor of the Earth Kingdom capitol, has a bunch of kids talking about the war, which he keeps quiet behind city walls. Also, they have crucial intel on the Fire Nation. You'd Expect: Long Feng to give them the "no war within the walls" speech, then get the intel. The Earth Kingdom wins the war, leaving Long Feng as Supreme Dictator of the entire world. Instead: He ignores them entirely for no apparent reason. Hey, Long Feng, your whole "no war" thing only lasts as long as you can repel the Fire Nation, and surely Ba Sing Se has an astronomer on hand who knows that in a few months, the Firebenders are going to have a major power up.
And of course, the defining idiocy of the series comes from, again, Admiral Zhao during the first season finale. Zhao is invading the Northern Water Tribe in order to stop Aang from learning waterbending. Zhao is also aware that the Ocean and Moon spirits (the entities that power waterbending) are in the heart of the Tribe's city in the form of relatively harmless koi fish. Killing the Moon Spirit would destroy waterbending. You'd Expect: Zhao to realize that killing the moon is not only a mite bit overkill, but also would hurt the Fire Nation just as much as the rest of the world. So he'd think of some alternative that isn't nearly as stupid. Instead: Zhao kills the moon. Brilliant. And it's pretty clear that to him, the destruction of waterbending was a bonus. He just wanted to be able to brag that he killed the moon. Fortunately, Yue is able to undo his actions with a Heroic Sacrifice.
The newly minted Avatar Roku is celebrating his wedding, when he is approached by his best friend Fire Lord Sozin. Sozin tells Roku that it would be just swell to Take Over the World. You'd Expect: Roku to try and talk him out out it, to use the experience he gained in traveling the world for 12 years to explain why taking over the world is a bad idea, to take Sozin on a Diversity Tour of the world, to latch onto Sozin's arm and not let go until he is sure that one of the 5 most influential people in the world doesn't let go of his megalomaniac ideas. Instead: Roku blows Sozin off and tells him to just forget it. And when Sozin implores him to listen, Roku says that he doesn't want to hear anymore of this, leaving Sozin feeling betrayed by his best friend, bitter and isolated. This war really IS your fault, Roku.
Zuko does a few less than brilliant things, but consider the time he finds out that Aang in in Ba Sing Se, too, thanks to the "lost pet" flyer about Appa. You'd Expect: If Zuko really wants to take up chasing Aang again, he makes use of Aang's address on the flyer. Instead: He infiltrates the base of the local Secret Police, which is made up mostly of elite earthbenders, planning to steal a huge, flying furry monster which has horns, but no reason to be cooperative, what with having defended Aang against Zuko before. And Zuko has no idea whatsoever what he'd do if he actually got Appa out of there. It takes his Uncle coming after him and spelling out the flaws in his plan and how he's blindly following someone else's will to snap him out of it.
Iroh has just made it into the heavily-defended Earth Kingdom city of Ba Sing Se alongside a boatload of Earth Kingdom refugees, who would not look kindly upon a Fire National sneaking in with them. He has just gotten a cup of too-cool tea. You'd Expect: Iroh would just throw the tea away and wait to get something better, or simply drink it up and bear it. Instead: He uses firebending to heat the tea up, putting himself (and, by extension, his nephew) in danger of being lynched as Fire Nation spies. And, in fact, he does get noticed by Jet; while things don't exactly go as Jet planned, he still came very close to outing them as firebenders.
Wan Shi Tong has let the team into his library, one of the most sacred things he has. He is incredibly suspicious of them and all humans, since the last one burnt down a section of his library and used his knowledge not only for destruction and violence, but also to murder a fellow spirit. He tells them not to use the information for destructive purposes. However, Sokka finds some partly destroyed information which he can use to find a fire-nation weakness. You'd Expect: They'd use the utmost discretion, continue to check for his presence, not only out of respect for abusing the library, but also because Wan Shi Tong is a spirit (who dwarfs them), and the last angered spirit took on an entire armada with the help of the Avatar. Instead: Sokka quite loudly announces that with the information they can finally destroy the Fire Nation, and Wan Shi Tong is obviously right behind him.
Sequel series The Legend of Korra Episode 10: Lin is fighting the Equalists using a combination of earth and metal bending. Cue the arrival of the Lieutenant, a guy she's met before and knows for a fact has an electric current running through his kali sticks. You'd Expect: She would use earthbending to take him out then and there. Instead: She uses her cables to grab his sticks with metalbending, getting herself electrocuted in the process.
See also, the episode where the poor man's gundams were introduced. Having established that metalbending doesn't work and they're too strong to use cables on. You'd Expect: Lin and her police would use earthbending to at the very least upset the balance of the things and make it easier to pull them over. Instead: They continue to try and overpower the mechas with the cables and get their asses kicked because of it.
The Lieutenant walks in on Amon bloodbending Mako and Korra, proving he's been spouting lies to the Equalists. You'd Expect: The Lieutenant to alert the other Equalists of Amon's bloodbending statues, or at the very least perform a sneak attack on him. Instead: He charges head on, and gets pathetically thrown into a wall.