DNO said Tawke-21 and Tawke-22 were producing oil at a combined rate of 37,000 barrels per day.
Executive Chairman Bijan Mossavar-Rahmani said in a statement production from the Tawke field was close to what the company can deliver through existing pipeline infrastructure and by road.
"With the exceptional results from these latest wells ... we are on track to meet our ambitious deliverability goal of 200,000 barrels per day in 2014," he said Monday.
Genel Energy, led by former BP boss Tony Hayward, said in January an energy agreement between the semiautonomous Kurdistan Regional Government and Turkey, coupled with the completion of new pipeline infrastructure, should lead to a rise in oil exports from oil fields in the Kurdish region.
Kurdish oil issues are a source of contention for the central Iraqi government in Baghdad, which says unilateral KRG energy deals are illegal.
Turkey's official Anadolu News Agency reported Sunday about 1.35 million barrels of Kurdish oil are stuck at Turkish ports because of the Iraqi oil rivalry.
Turkish Energy Minister Taner Yildiz was quoted as saying the Kurdish region, and thereby the whole of Iraq, would benefit if the row were resolved.
"Baghdad will surely find a solution to this, for Iraq is a union," he said.