Although Iranian oil exports have recovered from Western sanctions that preceded the deal to restrict its nuclear development, exports from Venezuela have plummeted.
Brent crude futures (LCOc1) gained 85 cents to settle at $79.28 a barrel, while USA crude futures gained 18 cents to settle at $71.49 a barrel.
Brent, a crude reference in Europe, has reached 80.18 dollars per barrel, compared to 79.28 of Wednesday's closing. As for Opec, leading members have assured traders they have plenty of crude to make up for any losses from sanctioned Iran, though the cartel showed no signs of ramping up output.
The Organisation of the Petroleum Exporting Countries (Opec) has said that that it will not rethink its supply strategy even after oil reached $80 (£59) per barrel, labelling it a short-term spike rather than supply shortage.
"Longer-dated (crude) futures. remain in backwardation, driven by confidence in indefatigable USA shale producers", US firm Height Securities said in a note, although it warned that strong demand as well as looming disruptions due to renewed USA sanctions against Iran and falling output in Venezuela could soon start lifting the crude forward curve too.
The IEA said that the overall market balance was "continuing to tighten", and it lowered its estimate for 2018 global oil demand growth to 1.4 million barrels per day from its previous estimate of 1.5 million. "Supply concerns are top of mind after the United States left the Iran nuclear deal", said Norbert Rücker, head of macro and commodity research at Swiss bank, Julius Baer.
Oil stocks were expected to drop further as the peak summer driving season nears, offsetting increases in U.S. shale output, said analysts at Bernstein.More news: United States should honor Iranian nuclear deal commitment
US crude inventories dropped by 1.4 million barrels in the week to May 11, to 432.34 million barrels.
"While the sharp rise in United States production and rig count has raised questions on the sustainability of inventory draws through 2018, we believe that inventories will continue to draw as we enter the summer driving season in 2018", they said.
"The recent jump in oil prices will take its toll", said the IEA, which advises most major economies on energy policy.
The U.S. Commerce Department on Tuesday said consumer spending increased marginally last month, noting that higher gasoline prices were eating away at discretionary spending.
Analysts pointed out that the oil market remained relatively calm as there was no new catalyst on Thursday, but uncertainties remain due to US sanctions on Iran.
The IEA welcomed a statement from Opec kingpin Saudi Arabia last week saying it would work with other producers to limit the impact of supply shortfalls.