That doesn't sound correct. My understanding of the history is that Iran democratically elected a socialist who wanted to nationalize Iran's oil fields so they could keep the oil revenue inside their country instead of giving it away to BP and Exxon. The British orchestrated a coup to install the old monarchy (the Shah), who brought back the British extraction companies and harshly repressed Shia Muslims. Then in 1979 the Shia hardliners toppled the government in the Islamic Revolution which is where the current government originates from. The last real democratically elected president of Iran was the socialist one, Mohammad Mosaddegh.
No. Mossadegh was appointed by the Shah (who was still head of state), but his own autocratic actions such as dissolving parliament and giving himself autocratic powers pushed most political forces against him that a confrontation was inevitable. There is a reason the military stood back as he was disposed. Furthermore, the Shah did actually have the legal right to fire Mossadegh, when je ignored that the situation was already extra legal.
I don't know where you are reading history from but listening to random factoids rather than a comprehensive understanding is the worst way to do so.
The point GP made was that Mossadegh was not democratically elected. There hadn't been representative democratic support for Mossadegh. Mossadegh was installed by the Shah and Majles, stopped an election that wasn't going his way, and then tried to dissolve parliament to concentrate power with himself.
"Iranian people voted in their beloved leader, who was then toppled by the mastermind West" is a cartoonish level of geopolitical understanding by those who have read the first couple paragraphs of wikipedia
There has been a release of archived official US diplomatic communications a few years ago that paint a rather mixed picture where the embassy was convinced the uprising had failed and the next days there were surprising marches in support of the shah that succeeded in the end.
To be fair, the entire chain of this thread is lacking any sources. Wikipedia at least contains sources, despite its relative inaccuracies and questionable authenticity of those sources. "in conflict with wikipedia" seems somewhat reasonable at this junction until someone rises above that bar.