* Crude oil rises this week after five weeks of decline
* U.S. output growing, rig count highest in over 3 yrs
* Weekly U.S. API inventory data out on Tuesday (Adds comment, updates prices, changes dateline from SINGAPORE)
LONDON, June 27 (Reuters) - Oil prices rose for a fourth consecutive session on Tuesday boosted by a weaker dollar and investors covering short positions, but worries over persistent oversupply capped gains.
Brent crude futures, the international benchmark for oil prices, had gained 25 cents to $46.08 per barrel by 0901 GMT.
U.S. West Texas Intermediate (WTI) crude futures were up 23 cents at $43.61 per barrel.
The gains mean the market is up slightly so far this week, after spending much of the last month in negative territory.
The dollar fell 0.1 percent against a basket of six major currencies, propping up oil, ahead of a speech by U.S. Federal Reserve Chair Janet Yellen.
"Short-term financial investors also significantly scaled back their net long positions in Brent on the ICE last week...and find themselves at their lowest level in a year and a half," Commerzbank said in a research note.
"Short positions have soared to a new record high, having increased more than four-fold since the beginning of the year."
The Organization of the Petroleum Exporting Countries and its partners have been trying to reduce a global crude glut with production cuts. OPEC nations and 11 other exporters agreed in May to extend cuts of 1.8 million barrels per day (bpd) until March 2018.
Despite the cuts, which started in January, markets remain well supplied due to rising output elsewhere.
OPEC members Nigeria and Libya are exempt from the cuts and have raised production. OPEC member Iran was also allowed a small increase to recover market share lost under Western sanctions over its nuclear programme.
Libya's oil production rose to 935,000 bpd, up from 885,000 bpd last week, a Libyan oil source said on Tuesday. Meanwhile Nigeria's projected August exports are at their highest since March 2016 at around 2 million bpd.
Adding to the bearish supply outlook is an apparent glut in U.S. East Coast product supplies.
Colonial Pipeline Co said that demand on its main distillate line fell below capacity for the first time in nearly six years as the East Coast remained awash in fuel.
Meanwhile, U.S. shale oil output has risen about 10 percent since last year to 9.4 million bpd <C-OUT-T-EIA>, with the number of U.S. oil rigs in operation at the highest in more than three years.
"Traders are also looking ahead to the EIA Energy Conference in Washington, where U.S. shale oil producers are expected to give their view of current market conditions," ANZ bank said.
Analysts at Bank of America-Merrill Lynch said demand was not growing quickly enough to absorb output, especially since imports in Asia are stuttering.
A fuel glut in China, a hangover from demonetisation in India, and an ageing, declining population in Japan are holding back crude oil demand growth in three of the world's top four oil buyers. (Additional reporting by Naveen Thukral in Singapore; editing by Susan Thomas)