There are several wars going on in Iraq, and it's important to keep them sorted out. The main battle is between Sunni Arabs (about 20 percent of the population), and the rest of the population (60 percent Shia Arabs, 20 percent Sunni Kurds and several other religious and ethnic minorities). The big battle is between Sunni and Shia, and has been for over a thousand years. While some 88 percent of the 1.3 billion Moslems on the planet are Sunni, about 11 percent are Shia. While most other Islamic sects just represent religious differences, many Shia believe they should be running the Islamic world, and that all Moslems should be Shia. The Sunnis disagree, often violently.
What has kept the Shia cause alive all these centuries is the fact that some 90 percent of Iranians are Shia. Iran (also known as Persia, or Parthia), has been the major power in the region for over 2,000 years. The Parthians were the one group the Romans could not defeat. While Persia was overrun by the initial wave of Islamic conquest, the Iranians soon developed their own distinct form of Islam. They became Shia. So did many others in the region. Today, Bahrain and Azerbaijan are two-thirds Shia. Iraq is 60 percent Shia. There are many countries with a Shia minority. In Lebanon, the Shia are about 35 percent of the population. In Saudi Arabia, the ten percent of the population that are Shia are concentrated in the eastern part of the kingdom, where the oil is. There are many other Moslem nations with Shia minorities, including Afghanistan, Pakistan, Kuwait, the United Arab Emirates and India (where about ten percent of the 145 million Moslems are Shia.)
While the Arabs take great pride in the fact that Islam originated, and spread, from Arabia, there has always been an uneasy feelings that the Iranians would one day take over. When the Arab Caliphate began falling apart a thousand years ago, Shia Iran became relatively stronger. But then a series of external events saved the Arabs from Iranian domination. The Mongols smashed the Iranians up real good. A few centuries later, as the Iranians recovered, the Turks moved in to protect the Arabs. When the Turkish empire fell apart a century ago, Europeans and Americans arrived to keep the Iranians from taking care of their "Arab problem".
After World War II, all seemed well in the Persian Gulf. The Iranians were ruled by a monarchy, which recognized Arab control of Islam's holy places in Saudi Arabia. Then, in the late 1970s, the Iranian monarchy was overthrown. The revolt was supported by the Shia clergy, which was always more politically active, and better organized, than the Sunni clergy. The Iranians were going to establish a democracy, but then Iraq (led by Saddam Hussein) invaded Iran in 1980, hoping to grab some territory while the Iranians were disorganized from their revolution. That changed everything. The Iranians got organized, the Shia clergy took over and democracy got put on hold. Iran turned into a theocracy, run by the senior clergy. Worse yet, the clergy called for a world wide religious revolution. The world was to be converted to Islam, Shia Islam. But because 88 percent of Moslems are Sunni, the Iranian religious revolution didn't get very far. It wasn't for want of trying. The Iranian Shia sent money and guns to Shia revolutionaries all over the region. This caused some noise, and death, but never really gained much traction. The Sunnis fought back. The hard core Sunni clerics had always considered the Shia to be heretics, and this business of Shia religious revolution just made the Sunni fanatics madder.
Even before the radical Shia clergy took over in Iran, radical Sunni clergy were preaching Islamic world conquest. That slowly grew, until it became al Qaeda, and other like-minded groups, in the 1990s. But it got worse. Even before the "Islamic Republic of Iran" appeared in the 1980s, Saudi Arabia was funding religious schools all over the Islamic world, and encouraging the persecution of Shias. But once the Iranian theocracy got established, the Saudis, and other wealthy Sunni Arabs in the Persian Gulf, were funding Sunni radicalism, and encouraging anti-Shia violence. The Shia must not be allowed to spread their heretical teachings. This battle has largely been ignored in the West, but it has been going on for decades, and thousands die each year because of it.
As bad as Saddam Hussein was, the Sunni world saw him as a their defender against Shia Iran. When Saddam fell, Sunnis, especially Sunni Arabs in the Persian Gulf, saw Iran taking over Iraq, because 60 percent of Iraqis are Shia, and then taking over the Persian Gulf. While the West just saw Saddam as a nasty tyrant, the Sunni world saw him as someone who knew how to handle Shia scum. Again, this aspect of Saddam's popularity in the Moslem world went largely unreported.
But the Shia angle is key. A major component of al Qaeda's attraction is its willingness to go after Shia. This is why Iran never provided much support for al Qaeda. That Iran provided any at all merely demonstrates how eager rabid Iranian religious radicals were to strike a blow at the unbelievers (the West). But most Iranians hate al Qaeda, and Sunni radicals in general. Again, it doesn't get reported much in the West, but in Iran, and Shia areas elsewhere, the latest terrorist attack against Shia anywhere, is always big news.
Can the Iranians "control" the Iraqi Shia, and form a more powerful Shia alliance that could threaten Sunni control of the Persian Gulf? As a practical matter, no. For one thing, most Shia Arabs are Arabs first and Shia second. A few percent of Iran's population are Arabs, and they get treated the way Iranians treat all Arabs; poorly. In response, the Iranian Arabs are unhappy with their government, and often violent when expressing that unhappiness.
Secondly, the Western nations are not going to allow Iran to build a Shia empire. It's simply not going to happen. Even most Iranians don't want any part of this world conquest fantasy, which is only embraced by a rabid religious minority within Iran. A more likely conflict is one within Iran, between the radical religious minority, and the majority of Iranians who are tired of being ruled by a clerical dictatorship.
Within Iraq, there are many Shia factions. The more determined ones (and there are more than one) are willing to use terror and violence to establish a religious dictatorship in Iraq. But they are only factions (mainly the Badr and Sadr groups). Iraq, as a whole, is not tolerant of more dictatorship. Iraqis have seen what a religious dictatorship has done to Iran, don't want Iranians telling them what to do, and don't want another Sunni Arab dictator either.
Establishing a working democracy in Iraq is a messy business. There are many factions, and some of them are willing to terrorize and kill to get their way. Americans have a hard time accepting that sort of behavior as normal, but it's normal in Iraq. Every democracy is different, and the Iraqi democracy is going to be very different.