ARBIL (AFP) Turkish forces pushed into northern Iraq on Sunday, killing four people, including a 15-year-old girl, as they hit back against hideouts of Kurdish rebels who killed 12 soldiers in the deadliest spell of violence in two years. The ground incursion into Iraq came after Turkish Prime Minister Recep Tayyip Erdogan vowed to fight the Kurdistan Workers Party (PKK) to the end and PKK spokesman Ahmed Denis hit back with a threat to attack cities across Turkey if the army did not halt its policy of confrontation. Iraqi Foreign Minister Hoshyar Zebari, himself a Kurd, criticised the unilateral Turkish action and called on Erdogans government to return to peaceful efforts to woo Turkeys large Kurdish minority away from violence. No country should resort to unilateral action. Unfortunately this has not been observed, Zebari told AFP, noting that a tripartite security commission - established to tackle Ankaras concerns - involving Iraq, Turkey and the United States, had been sidestepped. He noted that Turkish air attacks had been going on for months and were definitely a violation of Iraqi independence, sovereignty and good neighbourly relations. By morning, the troops had advanced 10km into Iraqi territory in the Qandil mountains where the rebels maintain a network of rear bases in their 26-year-old armed campaign for self-rule in southeastern Turkey, the Iraqi Kurdish security official said. Turkish troops were operating in the mountains north of the town of Sidikan in Arbil province, one of three that make up the Kurdish autonomous region of northern Iraq, the official said. Turkish fire killed a 15-year-old girl and wounded her mother and two-year-old brother in Khwakurq village, Sidikan district commissioner Ahmed Qader told AFP. As Turkish forces advanced into Iraqi territory during the night they killed another three people, a security official said, without specifying whether the dead were civilians or PKK fighters. Inside Turkey, the rebels kept up their attacks, killing one soldier and wounding another, bringing the militarys losses in the past two days to 12, Turkeys Anatolia news agency reported. The new casualties came as the Turkish Premier joined other dignitaries in the eastern city of Van to honour the 11 soldiers who died on Saturday - the bloodiest single day for the army since 2008. We are not going to slide into defeatism... we are going fight on to the end, Erdogan said at the televised ceremony. Last year, Erdogan had announced a new policy of boosting Kurdish freedoms and investment in the countrys impoverished southeast. The Iraqi Foreign Minister called for a return to the previous policy. This initiative is a wise one and needs to be embraced, enhanced and translated into action as the best solution for this age-old problem, Zebari told AFP in an interview. Zebari stressed that Baghdads relations with Ankara remained guided by mutual respect, adding: We do not condone or support any cross-border terrorist attacks by the PKK but, really (Turkeys air attacks) are not the way to handle this problem, and nonetheless no country should resort to unilateral action. Zebari said he feared Turkey was stepping up its incursions and air raids as the clock ticks down to an August 31 deadline for Washington to pull out 38,000 of its remaining 88,000 troops in Iraq. I personally believe the reason they are escalating these attacks now is to test the will of the Iraqi government, and also the American forces, as a prelude to the withdrawal of US combat forces in August, Zebari said. We are capable of filling the vacuum and we will not allow any other countries to step in to fill that vacuum.