An easier solution would be to tell the kids that cooties will grow in the walls if they don't turn on the fan. That'll do the trick if they're young enough. If mold appears, tell them it's the cooties breaking through.
Barring that, there is probably a solution to your query but I don't know what it is.

This is certainly possible, but probably not with off the shelf controls that go into your typical wall junction boxes. You might have to use some low-voltage relay controls and employ a creative, experienced electrician or an electrical engineer. At minimum you will need to get a hobbyist interested. Having the lights on a dimmer might complicate it considerably.

If I were starting to plan such a project, I would be looking at this as part of the project. I've used a couple of them and they are reliable.

http://www.electronickits.com/kit/complete/elec/ck1614.htm

Thanks! Well, my DH is a mechanical engineer, let's see if he wants to dig in into this kit :-)

As much as I love designing and building electronics, you could simply use an off the shelf occupancy sensor switch.

The switch would not tie to your lighting control but simply turn on the fan when someone enters the bathroom. When they leave it will turn the fan off after a preset amount of time.

Here's a link to a Lutron occupancy sensor on the Lowe's website as an example. The Lutron info states it can be set for a 1, 5, 15, or 30 minute timeout and will work with fans up to 3 amps.

Here is a link that might be useful: Lutron switch available at Lowe's.

If you're worried more about steam/condensation, this is perfect - it can be switched on manually, but also comes on automatically when it sense humidity above a certain level.

Even right next to the bathroom door, when it's ajar, it still is quite effective. If you forget to turn it on before your shower, it will come on not long after you start the shower. Sorry it's a Canadian link, but if you google the product name, you should be able to find it somewhere in an US-based store.

An occupancy-sensor switch would be good, but make sure you get one that's compatible with fan motors - one rated for lights only could cause all sorts of issues, including a house fire.

Home automation like Insteon might be another way to go.

Here is a link that might be useful: Dew Stop switch

I agree with the two previous posts. They are simple solutions. Before I saw these, I had another thought. If you want a simple timer, and can sacrifice dimmer operation for bi-level lighting, just put one light and the fan on one switch. A second switch can be set in the same box to turn on the rest of the lights but only when the switch that controls the fan and the first light is in the on position. In fact, the second set of lights could be put on a dimmer if you like.

Ionized: The old 555 timer kit. I actually have an Electronics Australia project kit the "Flextimer" sitting in my basement, ready to assemble. I can't remember what I bought the kit for.

Similar to the module you showed there, handy for lots of projects. OP will need some space to mount it, power supply and extra relays etc, but I'm sure it would work. Probably not really code, especially if it was mounted in the wall. Would be fun to set up, though.

Another thought: Microchip kids with RFID chips, put a reader near the door. As soon as sensor detects smelly kids, fan switches on.

Perhaps a modified gas detector could be used to detect elevated levels of methane during toilet use, and again switch on fan? I have to admit, I'm curious whether flatus would set one off. I'll have to experiment.

"Perhaps a modified gas detector could be used to detect elevated levels of methane during toilet use,"

The typical natural gas detector is not sensing methane. and there is not usually all that much generated by humans.

True. But natural gas is 95% methane.

I was being 100% facetious.

I don't know how accurate those detectors are, the one I had (Kidde Combo CO/NG detector) seemed to false alarm a lot. Somehow my laser printer would set it off.

I definitely know they consider Butane propellant (sprayed some aerosol cleaner near one) "gas" which is fair enough, it's equally dangerous.

If the nat gas detectors are not working on methane or the other simple unsaturated hydrocarbons fellow travelers, what are they working on, the mercaptan oderants?