Government collapse and civil disorder

I've a democracy in which the government has collapsed as a city was in civil disorder for a couple of turns. Which now means half of my civ is in disorder. I might be getting confused with a different version of Civ, but I thought that you were able to form a new government after a few turns, even if all those cities remained in disorder.

Or do you have to click round each city and change the conditions so they're not in disorder??

I guess you'll just have to wait the anarchy out, but I can't really tell you for certain if the cities that are in disorder will affect the length of the period. Theoretically, they shouldn't, as you're now in a different form of government, anarchy, so the dangers of perpetual disorder should be confined to A) improvements getting destroyed, and B) a small chance for some of your cities to do a culture flip if there's a sufficiently awesome AI player on the planet. But still, I've rarely experienced government collapse, since I only use democracy when I'm already too far ahead and can easily and painlessly deal with unhappy bums by pouring milk and honey over them with a 40-50% lux rate.

You have to address the cause for disorder.
Usually with Democracy the problem is having a unit outside the city. It's one of the major limiters to democracy in the game.
I've been stuck in a situation where I must disband a unit just to prevent the government from collapsing. If you go look at the city in disorder, the unit they're mad about will have a black icon behind it in the list of units that city supports. Find the unit and move it to the nearest city. Then use 'H' to set that city as its newest home (of course it needs to be able to support it).

This is a problem you need to be well-versed with if you want to deploy the best strategy for the game: Rush pyramids, and go early democracy. You won't be warring early, but you'll get a massive tech and population lead. You can switch to communism if you want to war for a bit.

Hope that helped