levelling up

I've ran two sessions of Apocalypse World now (one character-gen, and one full session), and the game's gone great. But, in the main session one character managed to roll against a highlighted stat four times and lapped Hx once, and is now looking to level up.

Is that how fast characters are meant to advance, or am I doing something wrong?

You're not doing anything wrong.  Characters are cool, powerful, and throw their weight around.  We're watching them as they go through very interesting times.

Kind of like Deadwood or Battlestar Galactica. 

Starbuck grows in those moments where her life is on the line and she's doing interesting stuff.  Sometimes that means she grows fast (like when she goes up against Scar).

With Ry. Sometimes it's that fast, sometimes slower. I mean, within any given game, there'll be fast-improving sessions and slow-improving sessions, fast-improving characters and slow-improving characters.

Make sure that you're playing equally with all the players, and asking everybody what they do (not telling them), and otherwise it's out of your hands. Your job isn't to control the rate at which people improve, so no need to worry about it!

That's great - thanks for the speedy reply. I just didn't want to run out of road by having a character reach some sort of maximum too quickly.

I guess, as the player herself told me, it gives them a strong incentive to be protagonists in the story.

Now I just need the rest to notice that :)

Another thing to do is be aware of what stats are highlighted. If I have a PC who's zooming ahead of the rest, it's usually because they have highlights in their strong stats, and they are rolling on their strong stats often. So maybe that's fine, and no big thing, and maybe I'll highlight something else next session, to see a different side of their character and give things time to even out.

This is related, but only a little bit.

I find when I have "cool" highlighted I am generally waiting for the MC to give me an opportunity to act under fire, like I keep saying I do more and more risky things and hope I get the chance, it's not so concrete. But when any other stat is highlighted I can usually just definitely do it.

Yeah, that too. I find I need to pay attention to what's highlighted, and then see if there's a reasonable option for that to happen.