With all of its questionable content and violence, Avatar: The Last Airbender is sure to have a lot of moments where the censors weren't guarding their posts.
Fans who toured the Avatar The Last Airbender studios returned claiming that the original designs of the swimsuits the girls wore in the episode "The Beach" were rather... naughty. It was a Take Our Word for It, but they were implied to be something that most anime wouldn't have done in a Beach Episode. Considering that the final designs that made it to air were actually fairly risque for what was nominally a children's cartoon (including highlighting Ty Lee's, ahem, character growth), these rumored original designs were probably sacrificial lambs to let them get away with the bikinis for Azula and Ty Lee.
In the "Winter Solstice Part 1: The Spirit World" When Zuko finds his uncle Iroh he sees him bathing in a hot spring. Zuko urges him to get out of the water so they don't lose Aang's track. When Iroh submits to it he gets out stretching much to Zuko's horror and disgust. He realizes Iroh isn't wearing anything! The only censor is Zuko's hand. Many people still wonder how they sunk a naked fat guy into the scene.
Zuko: "Uncle? We need to move on. We're closing in on the Avatar's trail and I don't want to lose him." Iroh: "You're tired, Prince Zuko. Why don't you join me in these hot springs and soak away your troubles?" Zuko: "My troubles cannot be soaked away. It's time to go!" Iroh: "You should take your teacher's advice and relax a little. The temperature's just right. I heated it myself. [He breathes steam through his nose.] Zuko: "Enough! We need to leave now. Get out of the water! Iroh: "Very well! [He gets up, exposing himself.] Zuko: "[Shielding himself.] On second thought, why don't you take another few minutes. But be back at the ship in a half-hour or I'm leaving without you!"
Zuko's and Mai's behavior in "Nightmares and Daydreams": Keep in mind Mai has an entire house to herself (the website states her family is still in Omashu), and Zuko got rid of the chaperones earlier in the episode. At one point, the camera cuts away while they're together at Mai's house late at night. When the show returns to their side of the story the next morning, Zuko is still there, lounging on her couch, that is in her bedroom, with a badly messed up head of hair.
This line from Nightmares and Daydreams:
Aang: "No, Firelord Ozai, you don't have your pants!"
Ozai: "My royal parts are showing!"
The episode "The Southern Raiders" became (in)famous among the fandom for what is a truly epic instance of this trope. Zuko bumps into a very flustered Suki on the way to Sokka's tent, and she hurriedly excuses herself. He walks into Sokka's tent to ask him a question and finds him pants-less and surrounded by flowers and candles. He even greets Zuko with a suggestive "Well helloooo..." before he realizes who it is. After a short talk, he rushes Zuko out and sticks his head out to call for Suki. And if there was any doubt, Sokka is shown the next morning fiddling with a flower necklace for no apparent reason except to indicate he got lei'd and Suki got deflowered.
That's a NORTHERN water tribe tradition. The Southern Water Tribe knows nothing about it. Katara just wore hers because it was a gift from her mother.
In "The Cave of Two Lovers" the gang meets a group of nomads. Their behavior is basically that of a stereotypical hippie, bags under the eyes and all. And their leader is named Chong.
The dance in "The Headband" could be regarded as G-Rated Sex. Starts out slow and measured... and winds up with Katara and Aang panting and sweating profusely.
Katara: Aang, everyone's watching us... Aang: Don't worry about them. It's just you and me right now...
Meh. You could make a similar analogy about some real life dancing. It was a rather suggestive dance though.
In The Fortune-teller, Aang comes in beaming after eavesdropping on Katara's fortune
Sokka: Looks like someone had a pretty good bathroom break. Aang: Y-yeah... when I was in there... Sokka:I don't even want to know.
In Tales of Ba Sing Se Sokka is having a Haiku off with a leader of a poetry club which happens to be all girls. We get this exchange:
Sokka: Like I paddle my canoe, I'll paddle yours too!
This last line is punctuated by Sokka slapping his bottom.
There's quite a few instances of dubious beverages showing up in the show that probably wouldn't have flown in a lot of other shows. There's at least two bars in the series, one of which shows up twice and features June drinking something that probably isn't tea. Zuko's ship crew also drinks something out of tankard-looking cups. Toph's mother and Earthbending master are briefly seen drinking what appears to be a type of Asian hard liquor called bai jiu. And then there's the "cactus juice," which is even worse since it's the Avaverse's version of peyote (an illegal hallucinogen) and not just alcohol.
In "Bitter Work," Aang has to learn how to stand up to things like an Earthbender. Toph ends up going through his things and stealing his bag of nuts and his staff, which she uses to crack the nuts. Only in the end he has the balls to demand them back.
While relaxing on Ember Island, Katara announces that she has a surprise. Toph quickly replies "I knew you had a secret thing with Haru!"
"The Drill": a goopy slurry of rocks exploding out of a rod-shaped drill penetrating a hole in Ba Sing Se's wall looks like a more adult liquid is splooshing out—repeatedly. And considering it splatters all the way to the top of the wall and many of the characters get coated in it thoroughly—Sokka's even brushing it off his tongue... yeah... you'll want the Brain Bleach on hand.
In The Earth King there is a scene where Zuko dreams about a serpentine dragon encircling him closely and whispering in his ear with Azula's voice, in a tone that can only be described as extremely seductive. It's exact words are "It's getting late, are you planning to retire soon, my lord?...Relax, Fire Lord Zuko. Just let go. Give in to it." The Incest "Subtext" in that scene gets pretty close to just being "text", and would be pretty blatantly sexual even if the voice belonged to someone else.
I think that was more intended to represent Azula "seducing" him to The Dark Side, with the other giant snake representing Iroh trying to pull him away from it.
Not entirely certain about this one, but in the episode, "The Warriors of Kyoshi", when Katara ceases to sew Sokka's pants back together for his rather chauvinistic remarks, he complains and immediately asks for her to resume the sewing. Aang responds with: "Don't worry Sokka. Where we're going, you won't need any pants!" Where did they go? An island with a bunch of women warriors... Hm...
Refute: Aang was unaware of the Kyoshi people; they were going there to ride giant fish. Not a good idea to be wearing pants while you do that.