Last Iranian Shah came to power thanks to Shia Clerics who control the Iran today. The same clerics took lead role in 1953 anti Mosaddegh coup d'etat. Shah has later chosen to ignore those clerics, and they overthrown him, too, which happened in 1979. Mosaddegh was overthrown by same clerics, because he was a communist and that is perfectly understandable. Another factor which contributed to Shah's overthrow was his terminal illness. His sons were too young to successfully inherit the throne, the oldest was only 19 in 1979. Shah has upset the clerics by pushing western lifestyle and secularism to some extent. I have never been to Iran, but my guess is that significant portion of population supported it and equally significant opposed it, supporting traditional values. Like in majority of countries in the world. Why would Iran be any different ?

I have used an awkward expression. Shah was a monarch before coup against Mosaddegh, but he didn't really run the country. Mosaddegh, as prime minister, did.