>How can she be the Avatar if she can be so easily defeated?
>Even the Avatar state was barely enough!

I found myself watching that battle closely because we saw on several occasions in the last series the folly of tempting the Avatar State. I was particularly comparing it to the Aang-Ozai battle which, once Aang hit the Avatar State, was mainly a hilarious montage of Ozai fleeing for his ever-loving life.

Korra and Zaheer's battle was very similar in the beginning, though without the "I dun fuked up now" look that was all over Ozai's face. There are some important differences though:

While Korra has had more time to properly train the four elements than Aang, hers has been school training where Aang's was real life/practical/combat training. Not dismissing Korra's innate passion and aggression, at the time of the Ozai fight and for the purposes of combat, Aang might have been a little better master of the elements than Korra (except fire for which he had the benefit of Sozin's Comet to make up for his lack of fire mastery).

More importantly, Aang had a much better connection to his spirit than Korra, likely making his Avatar State a lot more powerful than hers. All Avatar State's probably aren't made equal.

Most importantly, Korra was fighting poisoned. And not I'm going to be sick poisoned, but I'm going to die poisoned. She was fighting Zaheer while dying, so we probably weren't seeing her best Avatar State.

So I wouldn't say that the Avatar State was barely enough because I think it was a severely handicapped Avatar State.

As for the Tenzin/Zaheer fight, I said before, the whole time I was rooting for Tenzin to kick his ass, and the fight turned out to be close to a stalemate. The thing about Tenzin, from my reckoning, is that while he's an Airbending Master, that almost seems to be partially by default. Not saying that he doesn't deserve the title, but he's no Aang. I mean, part of the character's arc has been coming to terms with the fact that he's not as good as his father (in lots of ways) and that that doesn't disqualify him as worthy of being an Airbending Master and his father's son. Think about it, Aang, for several years at least and the totality of the time it took him to be an Airbending master (as a child, he was clearly a prodigy), had the benefit of many Air Nomads, Gurus, and a generally more spiritual world to learn from. Tenzin only had Aang. So it may not be fair to compare Tenzin's mastery to the superhero version of an Airbender (Aang).

Now Zaheer basically came out of the Harmonic Convergence a full fledged, old school Airbending Master. The show doesn't even attempt to explain how, but it seems pretty clear to me that he is. Obviously not everyone who gained Airbending started at the same level. Kai was much superior to Bumi. So the show establishes the pattern, at least, that not everyone starts as a complete novice Airbender. Beyond that, it's speculation.

You can speculate that the considerable focus, discipline, spirituality (projecting into the spirit realm isn't an Airbender ability), and sense of purpose that Zaheer had before Harmonic Convergence translated into his Airbending mastery. But I think clearly Zaheer is some kind of unqualified badass. Think about it, his crew was an armless waterbender, a combustionbender, a lavabender, and a spiritbender (Unalaq). He headed up a pretty unique and powerful group. What could he do before he got Airbending that earned their respect, allowed him to hold his own in that company, and got him locked away at the top of a mountain? I'm curious to know.