To wit, Abdullah Gul served as prime minister until his party's leader (Erdogan) became eligible to be prime minister. Yasar Yakis was Gul's foreign minister. Yakis stated that if Turkey had a legal claim to Iraqi oil fields AND they could prove it before the world, yes they would claim the oil. Turkish scholars responded that the question had been looked at and there is no legal claims to the oil. Turkey is owed a few million dollars dating back to an agreement decades ago, however. http://www.globalpolicy.org/security/oil/2003/0107turkey.htm
The Kurd question in further complicated by the fact that there are Kurd factions including PUK and KDP who battled one another in the 1990s. Turkey's biggest concern is the Marxist Kurdistan Workers party (PKK/KADEK). PKK attacks on the Republic of Turkey in the 1980s and 1990s cost an estimated 30,000 lives of Turkish citizens. Turkey defeated the PKK but many of them found refuge in a northern Iraq zone created by the allies after Gulf War I. The zone was for Kurd refugees but PKK also fled there. I believe THAT is what Turkey is most worried about.
I hope no one minds my comments here.