This is completely wrong and is historical revisionism. Obama ex-adviser himself wrote a book after he read the classified CIA files and he says in his book that it was completely a coup led by the CIA and the MI6. There is a lot more on the book and I recommend people who can get it to read it.
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>
The plan would consist in bribing journalists,
editors, Islamic preachers, and opinion leaders to undermine
the government and fan the flames of distrust. Stories
would be planted “to show Mosaddeq as a Communist
collaborator and as a fanatic,” and thugs would be hired to
organize protests and create chaos and instability on the
streets.
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> The shah would fire Mosaddeq, who was bound to
resist, creating a pretext for his arrest, his ouster, and the
restoration of the shah’s unquestioned rule.
On July 19, Roosevelt arrived in Iran and began to
implement the plan, paying agents to spread chaos on the
streets of Tehran and bribing members of parliament to
abandon the government.
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>The cautious, young shah—aware
of Mosaddeq’s popularity—was reluctant to approve the
action, but he was eventually brought on board, mostly due
to threats from Roosevelt but also in part by his sister
Ashraf, who was enlisted by the CIA to help persuade him.
Still dealing with the British-led global oil boycott, Mosaddeq
begged for assistance from the United States, but
Eisenhower declined to help, by then convinced that the
only hope was to squeeze Iran financially to prepare the
ground for the coup.
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So no it's not a conspiracy theory of islamists, it is in the historical record and the US government itself doesn't deny it/