Since Katara decided to abscond with so many episodes that should have been about other people, I decided to give Katara her own section in this guide. This isn't so much about Katara's personality as about what just happened in the last episode. Feel free to skip this if you're not interested.
The major problem with Katara is that she is predicated on a fundamental contradiction. She is introduced as a nascent waterbender, from a tribe who's waterbenders have all be captured and/or killed. She's "The Last Waterbender" of the Southern Water Tribe.
At the same time, her ultimate purpose in the series's plot is to be Aang's waterbending teacher. Aang, being a traveling Avatar, will need to have his teachers be mobile as well. They can't just be people he meets somewhere and occasionally gets instruction from; they have to be actual main characters. Gaang members.
These two things are somewhat mutually exclusive. You cannot credibly have an inexperienced waterbender be Aang's primary waterbending teacher. Therefore, to preserve both, you must have the inexperienced waterbender become experienced.
But how do you do that, without having Aang learn waterbending from whomever Katara is? Obviously, Katara must learn waterbending faster than Aang.
And you know what? I would probably have been willing to accept that. She is a natural waterbender who's been trying stuff on her own for some time. They even had that scene in the very first episode when she accidentally destroys an iceberg. Granted, that only showed strength, and as any martial arts practitioner will tell you, strength isn't everything. But it is something.
More importantly, Aang is shown to be irresponsible and not particularly disciplined. I could certainly see him not being particularly dedicated to learning waterbending, even with Roku's clock on the series. And that would mean that Katara, who really wants to learn it, would probably advance faster.
Unfortunately, this possibility was sabotaged thanks to The Waterbending Scroll. When I said that Aang learned waterbending instantly, I wasn't kidding. She showed him a move, and he immediately is able to perform it and do so even better than Katara. This even happened after they got the scroll; Aang learned from the scroll much faster than Katara.
At that point, it now becomes impossible to believe that Katara could learn waterbending faster than him. Even if he wasn't really paying attention most of the time, he picks up bending so quickly that he wouldn't need more than a lesson or two to learn any particular facet of waterbending.
Now, that being said, it's still theoretically salvageable. I am in a somewhat interesting position watching this show, because I happen to be adept in an arcane field myself (programming). So I have some understanding of how arcane fields work as you get deeper and deeper into them. You may initially progress quickly, but you then hit something that really stumps you or simply goes against whatever gave you that initial boost.
Being a fast typist alone doesn't make you a good programmer. Being good with languages alone doesn't make you a good programmer. These things can advance you quickly above neophyte programmers, but once you start getting into the deep magic, you had better know what all that stuff you did actually meant. Because if you don't, you're screwed.
The problem is this: if that's the direction they wanted to go in, they are obligated to show it. It means you have to show, in the context of the series, what waterbending is like and why Aang got blocked at some point. And that's generally why you don't see people going this route in writing. It's not a good idea. It requires you to talk about a fictional arcane subject in more detail than you probably wanted to and in more detail than the viewer cares about.
Or just not write The Waterbending Scroll. Everything was going reasonably well up until that point. But once you saddle the antithetical premises of Katara being both Aang's teacher and a poor waterbender, with the fact that she's also a slow learner, you basically throw away your one get-out-of-jail-free card.
Oh, but they try to make it legitimate. One of the first sentences out of Pakku's mouth in the very next episode is a line that desperately tries to get the audience to buy that Katara suddenly became awesome while Aang still sucks. In direct contradiction of what they said in The Waterbending Scroll.
It doesn't work. And it shows that they recognized their mistake with The Waterbending Scroll, but not before that episode was finished. So one wonders: if this problem is the result of that episode, why did they write it that way in that episode to begin with?
Because they had to. One of the primary purposes of The Waterbending Scroll was to help plausibly explain how Aang could learn 3 martial arts forms in about 8 months. Which basically shows that the whole idea was inherently contradictory to begin with: Aang has to be able to learn faster than Katara, since him learning the bending forms is the driving force of the show.
And it should also be pointed out that, from the beginning of the next episode forward, Katara is a waterbending master. No really; at the end of the upcoming two-parter, Pakku calls her a master. Let's consider this.
Time in the Avatar-verse is generally played loosely, a common tactic among writers who aren't able to plan everything out perfectly, or those who simply want to preserve the freedom to change things as needed. Even so, we know something about time here.
We met Roku on the day of the Winter Solstice, which happens in late December. And, in episode 8 of Season 2, it is important to the plot that it is springtime.
At best, if that is late spring (even though the episode strongly suggests that it's early spring, not late), this gives Katara about 5 months to train. And that doesn't include the time that the first 7 episodes of Season 2 take up, when Katara is no longer training. The Gaang goes to many places, and that takes quite a bit of time. The absolute best case you could legitimately say is that these seven episodes takes up a month, which leaves Katara with 4 months of training.
And even that means that the entire other half of the series (during which time Aang learns earthbending and firebending) all only take up about 4 months of time: from the end of spring to the end of summer when the comet arrives. The Summer Solstice, exactly half a year after the winter one, happens in episode 6 of Season 3.
This means that, in the most forgiving of estimates, this only leaves one month for all of the episodes between episode 2.8 and episode 3.6. We know that there are at least 4 weeks between 2.11 and 2.17. We also know that at least two weeks pass between 2.20 and 3.1. So that's simply not possible; the most forgiving of estimates just doesn't work out.
A more accurate estimate is maybe a few weeks of training. A month and a half, tops.
So no, I don't buy it. Unless Avatar-verse years are considerably longer than regular Earth years (in which case, it is their responsibility to inform the audience), there's really no way to do this.
So, what is my point? This is all a semi-obscure issue with the show and one character in particular. Why bring this up?
First, because many of the fans of this show hold this series up as an epic work of excellent writing, to be emulated by all. And this gaping plothole shows this to not be the case. That isn't to say that the series is bad, but perhaps people should reconsider their unmitigated praise for it.
More importantly, it plays into an argument I'm going to make in the introduction to season 2 (as does a lot of the things I've been talking about). Because the writers generally won't make these mistakes again. And it's important to note what these mistakes are.
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