Are you using a “real” MB program (not one “for kids”)? You can change the settings to increase the accuracy required to advance, and set goals for typing speed.
One thing I would say is to check if he is looking down at the keyboard when he types. My kids always want to look, but I keep reminding them that if they really want to improve their speed, they have to keep their eyes on the screen. It seems like it slows them down, but that’s how they will really pick up speed in the long run. Once they “buy into” that concept, they really begin to take off.
Finding real-life reasons to type is great, because it’s definitely about practice, like sheraz said, but if he is typing while composing of course he won’t type as fast. So typing from a printed text (like in Mavis) helps for just improving the skill itself.
If he is only typing 10-13wpm, I’d still definitely keep up with Mavis while incorporating some real-life needs for typing as well. Even emailing a friend or grandparent can be motivating practice. But I really, really feel like consistent use of a typing program, then transcription of a written text, will improve speed more than anything (as long as they keep those eyes on the screen, not the keys!) — because that’s how you get to the point of “muscle memory” or whatever you’d call it so you no longer have to think about where your fingers are moving. I keep my kids using some Mavis, at least in part, until they are at least well into the 30s or more.
But even then they definitely have to keep typing quite a bit once they get to that point. 🙂