Demonstrators have worried that the dramatic developments outside the US embassy would either steal their thunder or be mistaken for an extension of their own movement.
The attack on the embassy highlighted new strains in the US-Iraqi relationship, which officials from both countries have described to AFP as the "coldest" in years.
The United States led the 2003 invasion against then-dictator Saddam Hussein and has worked closely with Iraqi officials since.
But its influence has waned compared with that of Tehran, which has long and carefully crafted personal ties with Iraqi politicians and armed factions, even during Saddam's reign.
Both Washington and Tehran backed Iraqi security forces fighting the Islamic State (IS) group, but the two have been at loggerheads since the United States pulled out of the landmark nuclear deal with Iran in 2018.
Iraqi officials have feared that their country could be used as an arena for score-settling between Iran and the US.
"Before this episode, there was an agreement that in post-IS Iraq, the US and Iran don't attack each other directly," said Renad Mansour, an expert at the London-based Chatham House.
"That norm is being challenged now because Iran and its allies are in a bad spot. That is very destabilising, because they will seek to change the status quo."Follow us on Twitter and Instagram to stay connected