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UPDATE 4-Oil steadies as equities gain, but concern builds over supply

* World oil demand to grow by 1.62 million bpd in 2018 - OPEC

* Expects new supply to outpace demand-growth in 2018

* U.S. oil output hits record at 10.38 million bpd

* OPEC expects to lose market share as non-OPEC output soars (Updates with comment, refreshes prices; adds IEA report)

LONDON, March 15 (Reuters) - Oil prices were broadly steady on Thursday, supported by a pickup in equity markets, but held back by evidence that supply will overtake demand this year.

Brent crude futures were last down 13 cents on the day at $64.76 a barrel, while U.S. West Texas Intermediate (WTI) crude futures were up 1 cent at $60.97 a barrel.

Global oil demand is expected to pick up this year but supply is growing at a faster pace, leading to a rise in inventories in the first quarter of 2018, the International Energy Agency (IEA) said on Thursday.

"Oil and (the stock market) have been moving hand in hand ... which basically means oil is extremely sensitive to the growth outlook," SEB commodity strategist Bjarne Schieldrop said, adding he still expected demand growth to reach 1.8 million barrels per day (bpd) this year.

European equity markets were buoyed by strong earnings from several key financial firms, while U.S. stock index futures pointed to a pickup on Wall Street later.

The oil price has moved in sync with stocks uninterruptedly for the past 99 trading days, the longest such stretch in two years.

OPEC and several other non-OPEC producers led by Russia began cutting supply in January 2017 to erase a global glut of crude that had built up since 2014.

The IEA and OPEC both reported a modest rise in global inventory levels in January.

Looming over markets has been a relentless climb in U.S. crude output <C-OUT-T-EIA>, which hit another record last week by rising to 10.38 million bpd, up by more than 23 percent since mid-2016. Commercial crude inventories <C-STK-T-EIA> were up by 5 million barrels, at 430.93 million barrels.

"Surging U.S. output levels will continue to undermine OPEC's efforts for stronger oil prices," said Singapore-based brokerage Phillip Futures in a note on Thursday.

U.S. crude production, which has already overtaken that of top exporter Saudi Arabia, is expected to rise above 11 million bpd later this year, taking the top spot from Russia, according to the IEA.

The IEA believes non-OPEC supply, led by the United States, will grow by 1.8 million bpd this year, while demand will grow by about 1.5 million bpd.

The OPEC cuts combined with rising U.S. output mean OPEC is losing market share.

"In 2018, demand for OPEC crude is forecast at 32.6 million bpd, down by 0.2 million bpd from the previous assessment and 0.2 million bpd lower than a year earlier," OPEC said.

(Reporting by Henning Gloystein; Editing by Vyas Mohan and Mark Pottter)