Like any compelling character, the heroes of the Avatar universe are defined by their mistakes and failures just as much as they are their successes. Despite being the master of all four elements, each of them have fallen short of expectations in one regard or another.
Both avatars have made critical lapses in judgment that had the chance to (or explicitly did) impede upon the safety and wellbeing of those around them. Through finding their greatest and most galling mistakes, fans can better understand their imperfections as they fulfilled their inherited responsibilities.
10 Korra: Challenged Amon Alone & Was Humiliated For It
After Amon's attack on the pro bending tournament, Korra was enraged at his impudence. In order to quell the growing support for his Equalist movement, she openly challenged him to a duel, hoping that doing so would draw him of hiding and that she might dispose of him properly.
Once she had arrived at their determined location, the man was nowhere to be seen. However, he ambushed the Avatar just as she was preparing to go, demonstrating his power over her after patiently waiting to capitalize on his opportunity. Curiously, few held Korra accountable for the duel's outcome after the fact.
9 Aang: Tried To Hide Katara & Sokka From Their Own Parents
When it seemed as if Katara and Sokka would be reunited with their father, Aang was concerned about how this development might distract from their mission. As a result, he actively attempted to interfere with their reunion.
Worse yet, he would be discovered for his machinations, much to the shock and (rightful) offense of the siblings. It was only after Hakoda himself told them to return to the Avatar that they were willing to forgive their friend. His judgment was ultimately responsible for the world's salvation from Ozai.
8 Korra: Cheating On Bolin With Mako
Though initially infatuated with Mako, Korra sought out Bolin when the older brother seemed unattainable. This was a thinly-veiled ruse to garner the firebender's attention, a fact that was quickly pointed out before their unlikely kiss.
Not only was this a trespass of both of their boundaries (since the Avatar kissed Mako without his explicit consent and was cheating on Bolin), it was simply a selfish act unbecoming of an Avatar. However, the firebender would display similarly dubious behavior later in the series, when he attempted to keep his breakup with Korra a secret on account of her amnesia.
7 Aang: Abandoning Everything & Getting Frozen For A Century
The genocide of the Air Nomads was an understandably traumatic event for Aang. Everyone he knew— including his friends and mentor— had fallen before the depravity of the Fire Nation.
However, his decision to flee was incredibly selfish and not just to preserve his own safety. With his body encased in ice, he was unable to reincarnate, therefore preventing more intrepid avatars from assuming the mantle that he himself refused to fulfill. As a result, the Southern Water Tribe was brutally savaged, with the Northerners nearly facing a similar fate.
6 Korra: Merging With The Spirit World
Though the idea to merge with the Spirit World was initially Unalaq's, Korra decided it was ultimately in the best interest of both parties. This would soon prove to be demonstrably false and for a number of reasons.
Not only are the ways of the spirits and humans incompatible, their introduction wrecked havoc on Republic City. Further, the presence of spirit vines allowed Kuvira to create a devastating superweapon strong enough to hold entire nations hostage.
5 Aang: Sacrificing The Avatar State For Katara
Guru Pathik was responsible for helping to cultivate Aang in his journey to master the nature of the Avatar. Their final task was to fully unlock the power of the avatar state, only for the airbender to realize that Katara was in mortal peril.
Despite his teacher's warnings that abandoning his training would lock him out of his enhanced state forever, he still left the nomad's temple. Though he would mysteriously retain his strength regardless (proven through the final fight against Ozai), his willingness to put his love interest over the fate of the entire world is unbecoming of a hero.
4 Korra: Siding With Unalaq Over Her Own Father
While unable to explicitly prove Unalaq's evil, Korra's father, Tonraq, had warned her of the man's treacherous nature. Initially, she refused to listen, proving a willing pawn to the Northerner's machinations and opening one of the two spirit gates.
Once she began to follow his edicts, Unalaq declared martial law on the Southern Water Tribe, occupying it with his own military forces. Additionally, he nearly had Tonraq executed for a crime that Tonraq had no significant part in.
3 Aang: Overstressing About His Battle With Ozai
Though it may not have entirely been a conscious choice, Aang's overstressing of his impending fight against Ozai ironically hindered him. He lost sleep due to the coming battle and was unable to properly train as a result.
His lethargy was so conspicuous that both Katara and Toph couldn't help but to notice. Through the console of his friends and a reconciling with himself, he would eventually come to terms with what was inevitably fated to happen.
2 Korra: Losing Raava
Raava was the spirit of light that effectively made the avatars who they were. She greatly enhanced their strength and facilitated the connection with past lives. During Korra's fight against Unalaq, she was unable to defeat him, falling before his Vaatu augmented might instead and losing the spirit that permeated her soul.
While she would eventually rip Raava free of her uncle's body, she would never have access to her former selves again. This was particularly tragic since it was only through Aang's blessing that she was able to recover from Amon's bending stealing in the first book of the series.
1 Aang: Refusing To Kill Ozai
Aang's refusal to kill Ozai may have been inconsequential to the grand scheme of the narrative (since it was the end of the story with no room for the Phoenix King's escape), but it was still a staggeringly irresponsible decision. The man was a monarch surrounded by capable loyalists and a tactical genius intelligent enough to usurp the Fire Nation's throne.
Even without his bending, Ozai proved to be a genocidal menace deserving of justice beyond simply rotting in prison. Considering how easily characters have escaped from their cells in the Avatar universe before, allowing him to live was a nearsighted and selfish choice for Aang to placate his own conscience.