What is Turkey doing in Iraq?

As if we were short of tension in this region, a brand new crisis has erupted between Turkey and Iraq. 

Last week Baghdad bridled at Turkey's military presence in Bashiqa, a base north of Mosul where Turkey has been training Sunni fighters and Iraqi Kurdish Peshmerga. The U.S. has sided with the central government of Iraq on this longstanding issue between Ankara and Baghdad.

If the conditions were "normal," one could naturally ask what Turkey is doing in Iraq. If Iraq were able to prevent the penetration of terrorist organizations from within its territory to neighboring countries, and if foreign powers such as the U.S. and Iran were not swarming in Iraq, we could justifiably pose the question in the title above. But we cannot.

The anticipated anti-Islamic State of Iraq and the Levant (ISIL) operation to Mosul seems to be just around the corner. This upcoming offensive is quite critical as it aims to defeat ISIL in its capital in Iraq. But the operation's importance goes even further: The Mosul operation is a critical turning point that will define the sectarian (i.e. Sunni-Shia) equation and also the ethnic (i.e. Arab-Turkmen-Kurd) equilibrium in the country.
That is why Mosul is a very hot button for Baghdad, Erbil, PKK, Iran, the U.S. and Turkey.

Let's start with Iran. Iraq has come under the influence of Iran since the toppling of Saddam Hussein in 2003. Tehran has dominated the country mainly through its Shia militia. The so-called "Hashd al-Shaabi" Shiite militia, under Iran's domination, is all over Iraq on the grounds of fighting against ISIL. Now Iran also wants to be influential in the upcoming Mosul operation. Yet the U.S. and Turkey oppose this idea, worried that this might trigger another sectarian war in Mosul,...