BEIRUT, Lebanon — Iraq is, once again, deeply embroiled in crisis. For three years, the central government in Baghdad and the Kurdish region fought together to oust the Islamic State. Now, following the Sept. 25 referendum on independence for the region, they are pointing their guns at each other.

The dynamics in Iraq are far from simple, with intra-Kurdish rivalries; ethnic, sectarian and political divisions in Baghdad; and a war against the Islamic State barely in the rearview mirror. And yet too many people in Washington and elsewhere seem myopically focused on just one factor: Iran, which they view as controlling and dominating the situation in Iraq in pursuit of an ambitious, expansionist foreign policy. That’s far from the full story.

Since coming to power in 2014, Prime Minister Haider al-Abadi of Iraq has worked to push back against Iranian hegemony. Although he is (like the Iranian government) Shiite, he professes to be first and foremost an Iraqi nationalist. And he is certainly not an adherent of the Iranian government’s revolutionary ideology.

This doesn’t make Mr. Abadi unique. Most Iraqi Shiites likewise don’t want to see their country become Tehran’s puppet. The populist Shiite cleric Moktada al-Sadr now openly opposes Iranian dominance. Grand Ayatollah Ali al-Sistani, the Iraqi who is the global spiritual leader of all Shiites, has criticized Tehran’s interference, and in September, he refused to meet with a top Iranian cleric who had been dispatched by Iran’s supreme leader, Ayatollah Ali Khamenei. Even the Popular Mobilization Forces, a constellation of some 60 Shiite-dominated paramilitary groups in Iraq, are divided: Some are aligned with Iran, others oppose it.

To balance Iranian influence, Mr. Abadi has sought to build alliances with other local, regional and international players. He has visited Saudi Arabia, Iran’s regional nemesis, twice in recent months and has established strong ties with Washington.

The prime minister has also become increasingly popular with Iraq’s Sunnis, who are wary of Iran’s deep penetration into the Iraqi state since 2003 and now see Mr. Abadi as a conciliatory figure and a safeguard against too much Iranian influence.

The evolution of the fight against the Islamic State has revealed this balancing act. When the Islamic State swept across Iraq in the summer of 2014, Iran came to the rescue, quickly providing material and tactical support. Later, when Iraqi forces were advancing against the Islamic State, Mr. Abadi invited a United States-led coalition to join the fight — despite strong Iranian objections. And in more recent battles, Mr. Abadi has kept Iranian proxies back from the front lines.

This is some of the context that too many in Washington are ignoring right now as they view the tension between Baghdad and the Kurds as one piece in an Iranian gambit for control.

Iran is, of course, involved in its neighbor’s politics and battles. In fact, Iran is the strongest foreign actor in Iraq. And when it comes to Iraq, Tehran is always one step ahead of Washington and its allies because its relationships with groups from different sects and ethnicities give it a better understanding of the political terrain.

For instance, while American officials were left scrambling after the Kurdish Regional Government declined to postpone the referendum, Iran knew better — and was better prepared for the immediate aftermath of the vote. Iran is successful because it relies on powerful local allies, including friendly Shiite paramilitaries within the Popular Mobilization Forces. It also has a long-running relationship with the Patriotic Union of Kurdistan, one of the leading Kurdish parties.

This doesn’t translate to a desire to destroy Iraq. Iran, remembering the deadly war of the 1980s, wants to ensure that Iraq is not powerful. But its interests also include relative stability and a secure border. It can’t have the Islamic State too close; the Iraqi collapse in 2014 alarmed Tehran.

The fear in Washington has been the opposite: that Iran’s activities in Iraq are malign and destabilizing.

Since 2003, officials in Washington keep getting Iran wrong. Many are understandably still scarred by the 1979 hostage crisis or Iranian-supported Iraqi militias killing American soldiers in 2003. Or they are concerned about Iranian threats to Israel. When it comes to Iran in Iraq, emotion fogs reality, limiting understanding in Washington. Yet for Iran, when it comes to the United States in the Middle East, pragmatism guides policy, translating to more successes.

For the Iraqi government, Iran is a foreign actor just like the United States is. Iraqi leaders view Iranian officials in their country — including Maj. Gen. Qassim Suleimani, the commander of the Iranian Revolutionary Guard Corps’ elite Quds Force — as international military advises, just as they view American military advisers. Both are first looking out for their country’s national interests. The Iraqi government does not assign any ethical or moral superiority to one over the other, and it still needs both.

To finally get Iraq right, the United States must move past its obsessive denial based on a fear of Iranian influence, recognize that Tehran’s abilities are limited and appreciate the fine line that the government in Baghdad has to walk. Until then, the Iranophobes will be right about one thing: Iran is the smarter player in the region.