(CNN) Nowhere is the messy geopolitics of the fight against ISIS in Syria and Iraq more complicated than in the question of the Kurds.

The United States supports Kurdish groups in both Syria and Iraq as critical assets in the battle against the terror group.

But key US ally Turkey considers one of those groups to be terrorists -- and says it killed as many as 200 YPG fighters in airstrikes in al Bab, northern Syria in October.

In recent days, the Turkish government has also detained nearly a dozen pro-Kurdish lawmakers , in what Turkey's semi-official news agency Anadolu said was a terror investigation. Hundreds of people who turned out to protest against the arrests clashed with police in Istanbul on Saturday.

But across the border in Iraq, Turkey enjoys a close relationship with the semi-autonomous Kurdish Regional Government (KRG) in the north.

It even has at least 1,000 troops stationed in northern Iraq, training police and volunteers for anti-ISIS operations with the blessing of the KRG -- and much to the annoyance of Iraq's central government , another vital US ally.

Confused? Let's break it down.

Female YPG fighters walk to a checkpoint on the outskirts of the destroyed Syrian town of Kobane.

Who are the Kurds?

The Kurds are an ethnic group in the Middle East who have never had a nation state of their own, resulting in Kurdish nationalist movements across the region.

They make up a sizable minority in a number of Middle Eastern nations, comprising about 10% of the population in Syria, 18% in Turkey, 15-20% in Iraq, and nearly 10% in Iran.

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Kurdish fighters in northern Iraq, known as Peshmerga, and in northern Syria, known as the People's Protection Units or YPG, have proven some of the most effective fighting forces on the ground against ISIS.

Turkey is battling Kurdish militants

For decades, Turkey has been facing a violent insurgency from the Kurdistan Workers Party, or PKK -- a banned group that first took up arms against Turkey in 1984, seeking an independent state for the Kurdish minority concentrated in the southeast of the country.

Tens of thousands have been killed in the conflict, which resumed in earnest after a peace process collapsed in 2015.

The PKK is considered a terror group by Turkey, the United States and the European Union.

Turkey considers Syrian Kurdish fighters as terrorists, while its ally the US supports them

The main Kurdish militia force in northern Syria, which lies on Turkey's southern border, are the People's Protection Units, or YPG, often referred to as the armed wing of the Democratic Union Party (PYD), a Syrian Kurdish opposition party.

Turkey and others view the PYD and YPG as an offshoot of the PKK, indistinguishable from the separatist group banned in Turkey, says Fadi Hakura, a Turkey expert at Chatham House.

Kurdish YPG fighters fighting on the front line against ISIS.

As such, he said, members of the YPG are considered terrorists, and Ankara has worked to drive them from Syrian territory along its southern border, fearing that an entrenched Kurdish canton there would fuel momentum for an independent greater Kurdistan that could claim Turkish territory.

When Turkish tanks rolled into northern Syria in August, their military efforts were focused not just against ISIS, but also the YPG -- groups that were already fighting each other.

The United States, on the other hand, draws a distinction between the YPG and the banned PKK, providing significant support to the Syrian Kurdish group as a critical fighting force against ISIS and other Islamist groups in volatile northern Syria. This has complicated the relationship between Washington and Ankara, its NATO ally and key partner in the fight against ISIS.

For their part, the YPG/PYD acknowledge that they draw inspiration from imprisoned PKK leader Abdullah Ocalan, "but are careful not to link themselves openly" to the banned group, says Hakura.

But Turkey has a good relationship with Iraq's Kurds

Despite Turkey's tensions with Kurds at home and in Syria, it enjoys close economic and political ties with Kurdish authorities in northern Iraq, which lies to Turkey's southeast.

Photos: Mosul: Iraqi-led forces push into key city Members of the Iraqi federal police wave the country's flag as they celebrate in the Old City of Mosul on July 9, 2017. Iraq declared victory against ISIS forces in Mosul after a grueling monthslong campaign. The battle to reclaim Mosul, the last major ISIS stronghold in Iraq, has been underway since fall 2016. Hide Caption 1 of 73 Photos: Mosul: Iraqi-led forces push into key city This injured girl was found by Iraqi forces as they advanced against ISIS militants in the Old City of Mosul on Monday, July 3. She was carried away for medical assistance. Hide Caption 2 of 73 Photos: Mosul: Iraqi-led forces push into key city A suspected ISIS fighter is held in a basement while Iraqi forces continue to push for control of the Old City of Mosul on Monday, July 3. Hide Caption 3 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi Special Forces soldier exchanges fire with ISIS militants in the Old City on Friday, June 30. Hide Caption 4 of 73 Photos: Mosul: Iraqi-led forces push into key city A bomb explodes near the al-Nuri mosque complex on Thursday, June 29. Iraq's military has seized the remains of the Great Mosque of al-Nuri. Iraq and the United States have accused ISIS of blowing up the historic mosque. Hide Caption 5 of 73 Photos: Mosul: Iraqi-led forces push into key city Members of the Iraqi Federal Police hold a position as US-led coalition forces advance through the Old City on Wednesday, June 28. Hide Caption 6 of 73 Photos: Mosul: Iraqi-led forces push into key city The remnants of Mosul's ancient leaning minaret are seen in the Old City on Sunday, June 25. ISIS' claim that US warplanes were responsible for the destruction of the minaret is "1,000% false," US officials told CNN. Hide Caption 7 of 73 Photos: Mosul: Iraqi-led forces push into key city Two boys comfort each other after their home collapsed during fighting between Iraqi forces and militants in Mosul on Saturday, June 24. The boys, who are cousins, said some of their relatives were still under the rubble. Hide Caption 8 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi soldier helps transport a girl as residents flee their homes west of Mosul on Friday, May 26. Hide Caption 9 of 73 Photos: Mosul: Iraqi-led forces push into key city Displaced Iraqis make their way through rubble after evacuating their homes in a neighborhood of west Mosul on Wednesday, May 17. Hide Caption 10 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi man tries to extinguish a burning car during fighting in Mosul's western Rifai neighborhood on Tuesday, May 16. Hide Caption 11 of 73 Photos: Mosul: Iraqi-led forces push into key city A member of the Iraqi counterterrorism service secures a building as troops push toward Mosul's Al-Oraibi western district on Sunday, May 14. Hide Caption 12 of 73 Photos: Mosul: Iraqi-led forces push into key city A smoke cloud rises on the front line as the Iraqi Emergency Response Division advances in west Mosul on Saturday, May 6. Hide Caption 13 of 73 Photos: Mosul: Iraqi-led forces push into key city A wounded man is transported in western Mosul on Friday, April 21. Hide Caption 14 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi federal policeman smokes during a break from battle on Wednesday, April 12. Hide Caption 15 of 73 Photos: Mosul: Iraqi-led forces push into key city A member of Iraq's security forces stands guard in eastern Mosul as smoke rises from the ISIS-controlled western section of the city on Friday, April 7. Hide Caption 16 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqis visit a bath house on the southern outskirts of Mosul on Wednesday, April 5. Hide Caption 17 of 73 Photos: Mosul: Iraqi-led forces push into key city Flames billow from an explosion in Mosul during a clash between Iraqi forces and ISIS fighters on Sunday, March 5. Hide Caption 18 of 73 Photos: Mosul: Iraqi-led forces push into key city Families are forced to evacuate as Iraqi forces advance in western Mosul on Thursday, March 2. The number of internally displaced people has surged as the offensive effort has intensified. Hide Caption 19 of 73 Photos: Mosul: Iraqi-led forces push into key city Mosul residents cross a damaged bridge in the al-Sukkar neighborhood on Saturday, January 21. Hide Caption 20 of 73 Photos: Mosul: Iraqi-led forces push into key city French President Francois Hollande and French Defense Minister Jean-Yves Le Drian, right, view territory held by ISIS during a visit to a military outpost near Mosul on Monday, January 2. Hide Caption 21 of 73 Photos: Mosul: Iraqi-led forces push into key city A young girl takes part in a Christmas Day Mass at a church in the predominantly Christian town of Qaraqosh. The area's churches were heavily damaged by ISIS militants before the town was freed by Iraqi forces during the Mosul offensive. Hide Caption 22 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi Shiite fighters ride through a desert area near the village of Al-Boutha al-Sharqiyah, west of Mosul, on Friday, December 2. Hide Caption 23 of 73 Photos: Mosul: Iraqi-led forces push into key city Internally displaced Iraqis who fled the fighting in Mosul watch as a civilian drone films them at the al-Khazir camp on Thursday, December 1. Hide Caption 24 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi soldier searches a home for ISIS militants after Iraqi forces retook the village of Al-Qasr on Wednesday, November 30. Hide Caption 25 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi soldiers transport a comrade who was injured during a battle near the village of Haj Ali on Tuesday, November 29. Hide Caption 26 of 73 Photos: Mosul: Iraqi-led forces push into key city A man mourns during the funeral of four Iraqi paramilitary fighters who were killed in battles in the town of Tal Afar. Hide Caption 27 of 73 Photos: Mosul: Iraqi-led forces push into key city Displaced civilians return to the village of Tall Abtah on Friday, November 25, after Iraqi forces retook the village from ISIS. Hide Caption 28 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi civilians sit on the ground in Mosul on November 24. An Iraqi officer addressed the group, demanding to know the whereabouts of alleged ISIS militants who opened fire on troops a few days earlier. Hide Caption 29 of 73 Photos: Mosul: Iraqi-led forces push into key city An injured baby receives treatment at a field hospital in Mosul on November 15. Hide Caption 30 of 73 Photos: Mosul: Iraqi-led forces push into key city A woman cries Sunday, November 13, after seeing the St. Addai church that was damaged by ISIS fighters during their occupation of the Keramlis village. Hide Caption 31 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi special forces soldier prays next to a Humvee before troops pushed toward Mosul's Karkukli neighborhood on November 13. Hide Caption 32 of 73 Photos: Mosul: Iraqi-led forces push into key city A Kurdish Peshmerga fighter holds part of a defused bomb planted by ISIS militants in Bashiqa, Iraq, on Friday, November 11. Hide Caption 33 of 73 Photos: Mosul: Iraqi-led forces push into key city A member of Iraq's special forces guards two suspected ISIS fighters found hiding in a house in Mosul on November 11. Hide Caption 34 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi woman displaced by war holds her cat near a checkpoint in the Iraqi village of Shaqouli, east of Mosul, on November 10. Hide Caption 35 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi troops watch a broadcast of Donald Trump's acceptance speech in a house in Arbid, on the outskirts of Mosul, on Wednesday, November 9. Iraqi Prime Minister Haider al-Abadi congratulated Trump on his win and said he hoped for continued support in the war on ISIS. Hide Caption 36 of 73 Photos: Mosul: Iraqi-led forces push into key city US Marines install equipment at a coalition base in Qayyara on November 9. Hide Caption 37 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi forces member investigates a mass grave that was discovered after coalition forces recaptured the area of Hamam al-Alil on Monday, November 7. Hide Caption 38 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi children witness a man being interrogated by a member of the Iraqi army at a base next to the Al-Intissar neighborhood of Mosul on November 7. Hide Caption 39 of 73 Photos: Mosul: Iraqi-led forces push into key city A civilian man who fled the fighting trims his beard after reaching an Iraqi army position in Mosul on November 7. Hide Caption 40 of 73 Photos: Mosul: Iraqi-led forces push into key city Children play in debris created by an airstrike in Qayyara on Sunday, November 6. Hide Caption 41 of 73 Photos: Mosul: Iraqi-led forces push into key city Female members of the Freedom Party of Kurdistan sing as they hold a position near Bashiqa on November 6. Hide Caption 42 of 73 Photos: Mosul: Iraqi-led forces push into key city A baby is passed through a fence back to his mother at a refugee camp in the Khazir region on Saturday, November 5. Hide Caption 43 of 73 Photos: Mosul: Iraqi-led forces push into key city People line up to receive food at a refugee camp in the Khazir region on November 5. Thousands are taking refuge in camps set up for internally displaced people. Hide Caption 44 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi soldiers pass near a bridge destroyed in an airstrike in Qayyara on November 5. Hide Caption 45 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi soldiers patrol an alley on the outskirts of Mosul on Friday, November 4. Hide Caption 46 of 73 Photos: Mosul: Iraqi-led forces push into key city A suspected member of ISIS is detained at a checkpoint near Bartella, Iraq, on November 4. Hide Caption 47 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi families pack into a truck to be moved to camps on Thursday, November 3. Hide Caption 48 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi special forces soldier searches for the location of an ISIS sniper in Gogjali on November 1. Hide Caption 49 of 73 Photos: Mosul: Iraqi-led forces push into key city A man fleeing the village of Bazwaya carries a white flag as he arrives at a checkpoint on November 1. Hide Caption 50 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi soldier receives treatment after being injured during clashes with ISIS fighters near Bazwaya on October 31. Hide Caption 51 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi soldier navigates through a shattered windshield as coalition forces advance on Bazwaya on October 31. Hide Caption 52 of 73 Photos: Mosul: Iraqi-led forces push into key city Archbishop Yohanna Petros Mouche, center, performs Mass in the liberated town of Qaraqosh on Sunday, October 30. Hide Caption 53 of 73 Photos: Mosul: Iraqi-led forces push into key city Children play in a camp for internally displaced people near Kirkuk, Iraq, on October 30. More than 600 families from Tel Afar, a town west of Mosul, have been living in the camp since ISIS took control of the area in 2014. Hide Caption 54 of 73 Photos: Mosul: Iraqi-led forces push into key city Displaced families are seen on the road near Qayyara on Saturday, October 29. Hide Caption 55 of 73 Photos: Mosul: Iraqi-led forces push into key city U.S. military personnel take cover in a bunker after a mortar alarm was sounded at a coalition air base in Qayyara on Friday, October 28. Hide Caption 56 of 73 Photos: Mosul: Iraqi-led forces push into key city Women and children grieve over the grave of a family member at a Qayyara cemetery damaged by ISIS on October 27. Hide Caption 57 of 73 Photos: Mosul: Iraqi-led forces push into key city Residents of Qayyara wait for distribution of food and water rations on October 26. Local water sources have been contaminated by the burning oil and sulfur. Hide Caption 58 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraq's counterterrorism forces advance toward ISIS positions in Tob Zawa on October 25. Hide Caption 59 of 73 Photos: Mosul: Iraqi-led forces push into key city Soldiers give first aid to an injured boy in Tob Zawa on October 25. Hide Caption 60 of 73 Photos: Mosul: Iraqi-led forces push into key city Kurdish Peshmerga forces take positions as they start to move toward the Imam Reza and Tizxirab villages of the Bashiqa district on Sunday, October 23. Hide Caption 61 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi forces distribute fruit in the village of al-Khuwayn, south of Mosul, after recapturing it from ISIS on October 23. Hide Caption 62 of 73 Photos: Mosul: Iraqi-led forces push into key city Kurdish security forces detain a suspected member of ISIS in the eastern suburbs of Kirkuk on Saturday, October 22. Hide Caption 63 of 73 Photos: Mosul: Iraqi-led forces push into key city An Iraqi forces member helps a man push a car as they arrive at a refugee camp in Qayyara on October 22. Hide Caption 64 of 73 Photos: Mosul: Iraqi-led forces push into key city Spent bullet cartridges litter the street around the Jihad Hotel, where ISIS militants battled Iraqi security forces in Kirkuk on Friday, October 21. Hide Caption 65 of 73 Photos: Mosul: Iraqi-led forces push into key city Gen. Abdel Ghani al-Asadi, who leads Iraq's counterterrorism forces, sits in Bartella on October 21 after the town was reclaimed. Hide Caption 66 of 73 Photos: Mosul: Iraqi-led forces push into key city Peshmerga fighters look over a village during an assault near Bashiqa on Thursday, October 20. Hide Caption 67 of 73 Photos: Mosul: Iraqi-led forces push into key city Iraqi forces head toward the front lines near Qayyara on Tuesday, October 18. Hide Caption 68 of 73 Photos: Mosul: Iraqi-led forces push into key city A Peshmerga fighter peers up from an underground tunnel in the liberated town of Badana on October 18. ISIS fighters have built tunnels below residential streets to escape from airstrikes. Hide Caption 69 of 73 Photos: Mosul: Iraqi-led forces push into key city Kurdish security forces take up a position near ISIS-controlled villages on Monday, October 17. Hide Caption 70 of 73 Photos: Mosul: Iraqi-led forces push into key city Smoke rises from a suicide car bomb attack carried out by ISIS in the village of Bedene on October 17. Hide Caption 71 of 73 Photos: Mosul: Iraqi-led forces push into key city Members of the Iraqi coalition gather around a fire at Zardak mountain ahead of the offensive. Hide Caption 72 of 73 Photos: Mosul: Iraqi-led forces push into key city Peshmerga forces deploy in the dark near the village of Wardak early on October 17. Hide Caption 73 of 73

The Kurdistan Regional Government, a semi-autonomous administration which presides over the predominantly Kurdish region of northern Iraq -- with the might of the Peshmerga military, a key force in the battle for Mosul , behind it -- has developed a deepening relationship with Turkey in recent years, says Hakura.

Among other things, the KRG had proved an important export market for Turkish products, and a growing source of energy for Ankara.

And the KRG's nationalist agenda was viewed by Ankara as less of a threat to its interests than that of Syria's Kurds, says Hakura.

For one thing, there was more affinity among Kurds in Turkey for the PKK and PYD than there was for the KRG, he said.

"Turkey's feeling is that the KRG does not have territorial aspirations in Turkey itself, compared to the PKK and YPG," he said.

Peshmerga forces attack ISIS targets during an operation to retake Mosul.

Will Turkey's strike on Syrian Kurds affect relationship with KRG?

While both the KRG and the PKK harbor Kurdish nationalist aspirations, the relationship between the groups has waxed and waned over the years, says Hakura. In recent times, the relationship has diminished as the KRG's ties with Turkey have grown.

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"At times, they've enjoyed an alliance of convenience where their interests converged," said Hakura.

The two movements stem from quite different political traditions -- the PKK an anti-feudal, nationalist movement, the KRG operating according to a more tribal model of governance -- and there have been tensions between KRG President Massoud Barzani and Ocalan.

As such, Turkey's strikes on Kurdish militants in northern Syria last month were unlikely to affect Ankara's relationship with the KRG. Ankara says 160 to 200 Kurdish fighters were killed, although the HDP, a predominantly Kurdish political party in Turkey, says 14 people died.

"I doubt that the KRG will express any displeasure beyond rhetoric," said Hakura. "They will avoid any rupture with Ankara."

Massoud Barzani, the head of the Kurdistan Regional Government.

Turkey's military support for the KRG has angered another US ally

That relationship, between Ankara and the KRG, has proven another pressure point complicating anti-ISIS efforts in recent times, with Iraq -- another vital ally in the fight -- objecting to the presence of Turkish troops in northern Iraq.

The troops have been at a camp in Bashiqa, near Mosul, training police and volunteers for the fight against ISIS under an agreement with the KRG. Baghdad has objected to their presence, saying they are there without the central government's permission, but Turkey has insisted on the legitimacy of its stake in the future of the region, fueling tensions between the countries.

While the Peshmerga are a large part of the operation to drive ISIS from its stronghold in Mosul, northern Iraq, Baghdad is unhappy about the challenge to its authority presented by the Turkish troops' presence -- and ultimately is concerned about the independence aspirations of the KRG, says Hakura.

For Ankara, said Hakura, the troop presence is about expanding its influence in northern Iraq and helping to shape the outcome of the battle there -- an area that is heavily Sunni, like Turkey, and has a sizable community of Turkmen, a group ethnically related to the Turks.