what would have happened if the United States and its allies had invaded Iran in 2003, instead of invading Iraq, and how would you make it happen?
 
It would be even stupider than invading Iraq...
Especially as Iran (pre-Axis of Evil speech) was willing to work with the US to fight the Taliban in Afghanistan (up to and including basing rights for US planes!).
 
A complete and utter nightmare. Iran has three times three times the population of Iraq, and a more stable(ish) government. The people wouldn't be as welcoming of the U.S. as the Iraqis initially were. Add in the same incompetent leadership as during the Iraq War (Rumsfeld, Wolfowitz, and Bremer), and disaster will ensue.

To effectively police the country, the U.S. would need to provide 1.5 million troops (using the ration of 1 soldier for every 50 Iranians). At it's peak, the entire U.S. military was only 2 million troops. Additionally, the same incompetent leadership that I mentioned above would push for a lighter "footprint", and not provide the proper amount of troops to secure the country (not that we would have them to provide).
 
what would have happened if the United States and its allies had invaded Iran in 2003, instead of invading Iraq, and how would you make it happen?

I don't think it's possible to make that happen. For one thing Iran had nothing to do with 9-11 and was in fact opposed to the Taliban regime. Besides that, Bush and co knew that the Iraqi army had been gutted by years of sanctions, making Saddam's forces extremely weak, unfit for anything other than maintaining order internally. Iran, by contrast, has one of the largest militaries in the world and also fairly advanced. It would put up far more resistance than the Iraqis could ever have hoped to put up in 2003, and you've got a hostile population three times bigger than Iraq's. There's really nothing to gain by invading Iran and Bush & co knew this.
 
It would have been a disaster for the United States.

Iran is mountainous and has a population that is both larger and more homogeneous. They also have a history of resisting invaders.

The proxies in Lebanon and the rest of the world would be a disaster on their own.
 
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