every few turns the counter advances a little bit....
additionally there are random events where you are asked how your regime does certain things, this can further advance the counter or even reduce it again. you can find these events either by investigating places of interesst on the map or sometimes they come to you... (trying not to spoil too much)

suma sumarum you don't get pressured a lot by it, but it's a good reminder to stay focused

I've been playing a game for a good while now, and have been able to keep the counter very low. It's not pressuring me at all.

You can also research a spell that lets you reduce the counter directly, though it's expensive to develop and to cast.

The sort of choices you make in dealing with events, how you interact with the Sorcerer King himself (he's actually quite talkative), and the status of the elemental shards all affect the doomsday timer; with a slow-timer game it's easy to keep it down as long as the shards aren't getting wiped out left and right.

Thanks for all your answers, guys!

There's lots of mechanics that can be done that can decrease or increase the counter throughout the game.

There is also an option to set the speed to slow, normal, or fast when starting a game.