(Reuters) - More than 19,000 people have fled fighting raging in the east of South Sudan and sought refuge in neighboring countries since the start of the year, the United Nations said on Wednesday.

Almost 12,000 have entered Ethiopia from South Sudan's Jonglei state, where the army is trying to put down a rebellion.

Some 5,000 South Sudanese have crossed the border to Kenya, and another 2,500 to Uganda.

"The increase in numbers in Ethiopia in the past weeks has been particularly sharp, with around 4,000 new arrivals since 7 May," the UN said in a statement.

The violence has hampered government plans to search with the help of Western firms such as Total for oil in Jonglei, South Sudan's biggest state.

South Sudan's government has been struggling to assert itself across a vast and remote territory which won independence from Sudan almost two years ago.

This week the U.N. said it lacked sufficient troops and aircraft to protect the large numbers of civilians uprooted by the conflict during seasonal rains which turn much of the state into an impassable quagmire.

Earlier this month the army, an assortment of poorly-trained former guerillas from the 1983-2005 civil war, recaptured the rebel-held town of Boma near the Ethiopian border.

South Sudan's President Salva Kiir has renewed an offer of amnesty to help end the fighting, but also accused his own security forces of looting the town of Pibor, a remote settlement at the heart of the conflict.

(Editing by Ulf Laessing and Andrew Roche)