Initially two people were arrested after the incident - one was charged with aggravated assault and antoher for assault on a police officer.
Earlier Wednesday, police said two men were arrested for their role in the fracas.
The May 16 skirmish, which was captured on video, left nine protesters injured outside the Turkish ambassador's residence and further strained ties between the two countries.
The newspaper said the men had travelled to Washington to support the Turkish president, but it was unclear if they had a formal connection to his security detail. The Turkish Embassy claimed the demonstrators were "aggressively provoking Turkish-American citizens who had peacefully assembled to greet the president". They are among the most likely to face actual consequences.
Media reported that law enforcement officials plan to announce charges on June 15 against a dozen members of the Turkish president's security detail.
Rodman's N. Korea trip off to uncharacteristic low-key start
The BBC's Stephen Evans in Seoul says it remains to be seen if Mr Warmbier's illness is the direct result of brutality in prison. Any visit to North Korea by a high-profile American is a political minefield. "Otto should have been released from the start".
The four civilians accused have no such protection, however. He described the attack as "the type of offence that is extreme in nature".
The New York Times notes that this isn't the first time Erdogan's security detail has misbehaved on USA soil. Alpkenan Dereci, who was seen on video punching and shoving a protester, has not been charged with any offence. Two additional people were arrested at the scene on the day of the protests. However, the city's police chief said investigators had insufficient evidence to charge Erdogan in what he characterized as a "pretty brutal" attack.
"I condemn this attack, it was an affront to our values as Washingtonians and as Americans, and it was clear assault on the First Amendment", Bowser told reporters Thursday. It also declares that "U.S. respect for free speech requires USA officials to speak out against such incidents", a sad attempt at forcing Donald Trump, who hasn't publicly addressed the situation, into giving a shit.
Members of Congress from both parties have expressed their continued outrage over the incident, with senators John McCain, R-AZ and Claire McCaskill, D-MO both calling for the expulsion of Turkey's ambassador. Yildirim is awaiting extradition from New Jersey, the statement said. It occurred in broad daylight on a quiet Washington street soon after Erdogan met Trump at the White House. The spectacle of the bodyguards of a foreign autocrat attacking people exercising their free speech rights was unprecedented in recent memory. Turkey is also a key USA ally in its campaign against Islamic State militants, but the two nations have butted heads over their respective relationships with Kurdish fighters, some of whom the US has armed even though Turkey claims the groups are terrorist organizations.