Turkey is virtually completely Sunni Iran is largely Shia Syria...

Turkey is virtually completely Sunni
Iran is largely Shia
Syria is a Shia puppet with a Sunni majority

Iraq is 65% Shia and 35% Sunni with a Shia leadership

The Kurds are a third overlapping demographic - an ethnic one (without it's own truly independent state) that is mainly Islamic of which the majority are Sunni.

The majority of Iraq's Sunni population are Kurds.

The Kurds in the region represent roughly 14.5 million in Turkey, 6 million in Iran, about 5 to 6 million in Iraq, and less than 2 million in Syria.  Approximately 30 million Kurds spread across 4 adjoining countries.

The Kurdish muslims generally follow a more moderate (or less fanatical) brand of Islam.

A stateless group spanning the four adjoining countries in this power play.

A formal Kurdish state would be a negative for both Turkey and Iran, ultimately creating a potentially powerful and moderate barrier to their own conservative brands of Islam and the natural extension being their attempts at expansion and domination of the region.

The only axis I can see forming involving Iran (Syria, Iraq) and Turkey is one of short term convenience that does not afford any collective traction to the Kurds.

Islam 101 at play here - rinse repeat for personal/factional gain only.

In stating the above, it's only a very obvious macro viewpoint - there are no doubt many other (100-fold) strategic components and complexities that define the situation - although it is impossible to define in any single way.

Back to Iraq - it is a geographical trump card for whoever controls it - similarly Syria, as a gateway to the Mediterranean - hence Iran's (and Russia's) local geographic interest.