Is it fair to characterize the 1979 revolution as extremist without qualification? We find it extreme by our own standards, but by most accounts Khomeini was very popular in Iran.
Popular and extremist are, surprisingly, not mutually exclusive. I remember the Revolution pretty well and have talked to exiles of various stripes over the decades (including pre revolutionary exiles in France).
The royalist regime installed by the CIA in ‘53 replaced a popularly elected social democratic government. I won’t say the shah was ever popular, but there was not massive opposition at first. In part due to repression, and in part due to increased liberalization (e.g. rights for women), industrialization, and increased wealth (which I believe would have happened with the prior regime as well).
But those other factors don’t increase linearly or automatically and soon repression was the only way the Shah’s regime stayed in power.
To many people the Revolution seemed at first like a breath of fresh air. After all it overthrew the hated regime. This is the classic difficulty for all movements (from free software to regicide): different paths intersect and then diverge. Khomeini was indeed quite popular (though far from universally) but his followers were the largest (practically only iirc) organized opposition and so, like the Bolsheviks/Mensheviks (despite the name, the Bolsheviks were in the minority), they pushed the guys at the top of the hill off, and maintained their own balance up there.
Compare that to the French Revolution where there was no faction large enough to do this: they pushed out the Ancien Régime but then muddled around aimlessly and violently until someone was able to secure the high ground.
So Khomeini was “popular” in the sense that lots of people were favorable to his movement in the moment, but not “popular” in that the movement had a widespread base of support. Like the prior regime, after a short honeymoon they could not rule without repression.
Unfortunately there is no external opposition that could come in and restore democracy. So a complete overthrow of the current regime is quite unlikely.
I'm not so much talking about the revolution as I am the extremist groups that have developed and come into power since the revolution. It wasn't a one and done deal. Western intervention got the ball rolling for the development of radicals who were then able to work their way into positions of power.
