Genel said exports of oil by truck over the Turkish border began in January. It said it expects exports to reach as much as 20,000 barrels of oil per day this year.
Genel Chief Executive Officer Tony Hayward said the company started the year strong with "encouraging results" from oil fields in the Kurdish region of Iraq.
"As political momentum continues to build and the construction of independent regional infrastructure moves forward rapidly, it is evident that 2013 is set to be a highly significant year for both Genel and the Kurdistan Region's oil and gas industry," he said in a statement.
Genel said it's made significant progress on getting a pipeline to reach the Turkish border. Conversion of a pipeline in the Kurdish region from natural gas to oil could be finished as early as the fourth quarter of this year.
Political disputes between the semiautonomous Kurdish government and the central Iraqi government over oil laws are seen as impediments to Iraq's oil potential.
Export restrictions had been in place as a sign of growing frustration with the central Iraqi government over oil laws in the country.