The Big Book of One-Shots

Explanation of my Fangirly obsessiveness (Avatar in Fairytail)

In their world

Think about it, within the ATLA/LOK universe it's accepted that the Avatar is easily the most powerful person on the planet. Even if they're not well trained their sheer potential is terrifying and the amount of raw talent and power that has accumulated in them over the past 10 millennium makes it easier and easier for each new reincarnation to reach that potential. So the Avatar is scary enough in their own world.

But that effect is lessened by the knowledge people have regarding the Avatar. First of all, everyone knows that there is one. Always has been and always will be as far as they're concerned. People aren't like, "Oh my gosh you can bend multiple elements! What are you!". They're like, "Oh my gosh you're the Avatar! That's so cool!". They know what they are, what they can do, and they know that there's only one at a time. There's no worrying about, "Well, if you can do this, then who else can or might one day be able to?". Lastly, they know that it's the Avatar's duty to protect the world and keep it in balance.

Multi-classed magic

Now, enter the Avatar into a world that doesn't understand them, their lore, or even bending in general. {*cough* Fairytail *cough*} This is a world where manipulating the elements is accomplished through magic and is much rarer. 

Avatar: *tries to keep their head down and make a living in Fairytail*

Everyone else: Can't get their jaws off the floor because Avatar so-and-so can manipulate water AND ice.

Avatar: Seriously? Whatever. *heals someone*

Someone else: Holy, what the... *faints*

Avatar: Screw this, not even a normal bender could keep their head down in this place. What the heck. *starts using all four element freely*

Everyone else: *faints*

Avatar: *mutters* Lily livers.

Gray: You're clearly very powerful, so why can't I sense your magic?

Avatar: Uuuummm....

Now let's throw in some subsets:

Person: You can control metal too! I thought you said that you only knew 4 magic types! (can't believe I'm saying only four) Did you learn another one since then?

Avatar: It's literally another kind of Earth!

Person: Puppeteer magic?

Avatar: *turns red because it's technically illegal* Over half of the human body is made of water!

Person: Healing!

Avatar: Water control's the flow of energy in the body! Why can no one else do this?

Person: Is there anything you can't do?

Avatar: ... 

Avatar: Can I get back to you on that?

Immune to anti-magic spells

Now. let's think about the fact that this is what they can do without magic. Which makes the single best defense against powerful wizards, anti-wizard magic, useless against them.

Anti-wizard specialist: It doesn't matter how powerful your magic is, it fails you when used agai...

Avatar: *cuts them off by chucking rocks straight through their shield*

Why? Because "It's not magic, it's waterbending" - Katara

Wind wizard: *uses drain spell*

Avatar: *laughs* *Airbender slice!*

If you think that's scary enough you're wrong. There's more.

They're totally unpredictable.

They don't have to announce what they're doing or when they're doing it by shouting a spell and making a glowing magic circle. And they're not limited to one kind of attack. You can't put up a generalized defense mechanism and would have to learn a lot more about bending styles and general fighting body language to figure out what you even have to defend against. Non-wizards would probably actually have an advantage as they'd be more used to interpreting body signals instead of glowing circles and shouted commands. Even then, they wouldn't understand bending well enough to really know what would result from that movement.

Just for fun

Now just for fun, let's add my theory that bending is working with nature where as magic is beating it into submission. (my guess would be that dragon slayer magic is somewhere in the middle as it mimics the style of dragons) And nature doesn't like submitting. This would give benders in general the edge over wizards.

The frick frackin Avatar State

Lastly, let's throw in the Avatar State. Glowing eyes (and tattoos if applicable), can bend on a much larger scale than their own already impressive abilities, shaking elements ready to do their will (which by now means basically everything but dry wood, which gets blown around by the wind), hundreds of resonating voices speaking with one accord, one scary, no nonsense, calculated, very final accord.

Fairytail has officially found someone who scares them more than Erza.

I rest my case.

They don't even want to know about energy bending and whether or not a similar technique could be used on wizards. (probably wouldn't work, but they still don't know that the Avatar isn't a wizard, so...)

So that's what they can do with bending/slash Avatar stuff alone (by the way, this is all under the assumption of a fully realized Avatar). Depending on the person they may also be plenty capable without bending. And no one really even wants to think about what they could do if they learned actual magic.

That might be just a little bit too OP, even for them.