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UPDATE 4-Oil's price fall stalls despite supply glut

* Brent down 12 pct since OPEC-led production cut extension

* U.S. crude storage rises again this week

* Fuel demand slows despite peak U.S. driving season

* Floating storage in Asia remains common (Updates details and prices)

LONDON, June 9 (Reuters) - Oil prices steadied on Friday following steep falls earlier in the week, pressured by widespread evidence of a fuel glut despite efforts led by OPEC to tighten the market.

Brent crude oil was up 10 cents at $47.96 a barrel by 0920 GMT, around 12 percent below its opening level on May 25, when an OPEC promise to restrict production was extended into 2018. U.S. crude was 10 cents higher at $45.74.

The Organization of the Petroleum Exporting Countries and other big producers have agreed to pump almost 1.8 million barrels per day (bpd) less than they supplied at the end of last year, and hold output there until the first quarter of 2018.

But world markets are still awash with oil.

"The challenge OPEC is facing is bigger than anyone thought a few weeks ago," said Tamas Varga, analyst at London brokerage PVM Oil Associates.

U.S. data this week showed a surprise 3.3-million-barrel build in commercial crude oil stocks to 513.2 million. <C-STK-T-EIA>

Inventories of refined products were also up, despite the start of the peak-demand summer season.

"Crude oil prices are testing lows last seen in (the fourth quarter of) 2016," analysts at U.S. bank Jefferies wrote, pointing to the United States as the main pressure on prices.

U.S. refined oil product inventories are now back above 2016 levels and well above their five-year range, reflecting an unexpected slowdown in U.S. demand for gasoline and distillate fuels, Jefferies said.

Asian markets are also oversupplied, with traders putting excess crude into floating storage, an indicator of a glut.

The Brent forward curve shows a clear "contango" shape, with oil for use now at deep discounts to future prices.

Brent for January 2018 is worth around $1.50 a barrel more than Brent for August 2017, making it profitable for some traders to put oil into tankers and wait for a later sale.

Thomson Reuters Eikon shipping figures show at least 25 supertankers sitting in the Strait of Malacca and the Singapore Strait, holding unsold fuel.

That's similar amounts to May and April, indicating that even in Asia, with its strong demand growth, traders are struggling to clear inventories.

And more production is coming. Libya's 270,000-bpd Sharara oilfield has reopened after a workers' protest and should return to normal production within three days, the National Oil Corp said on Friday.

(Additional reporting by Henning Gloystein in Singapore; Editing by Dale Hudson)