"The violence must stop and he needs to step aside," Clinton told reporters in Paris after a meeting on Libya, where local strongman Moamer Kadhafi has already been forced from office.
A senior Syrian official, meanwhile, said in a contested video posted on YouTube that he has resigned in disgust at hundreds of killings and thousands of arrests by Assad's regime.
The Syrian Observatory for Human Rights said three people were killed and several more were wounded when security forces opened fire on protesters in the central region of Homs.
According to the Local Coordination Committees (LCC), which group activists on the ground, armoured cars entered the city of Homs itself.
The Britain-based Observatory also reported "one dead and five wounded in an assault by the army and security forces on the village of Al-Rama" in the northwestern province of Idlib, near the Turkish border.
And the LCC said a young man was killed by pro-regime militiamen in the eastern city of Deir Ezzor.
A girl of 10 was also reported to have died of wounds suffered late on Wednesday during a shooting near police headquarters at Deir Ezzor in eastern Syria as she rode in a taxi with her family.
The security forces went into action on Thursday after mass demonstrations late on Wednesday in several districts of Homs, the Observatory said.
It said security forces also opened fire to disperse demonstrations in the country's second city of Aleppo, in the north.
The United Nations says that more than 2,200 people have been killed since the beginning of near-daily protests across the country against Assad's regime in mid-March.
The head of the Syrian League for the Defence of Human Rights, Abdel Karim Rihawi, said dozens of people had been arrested on Wednesday at Qadam and Qabun in the Damascus region, and at Zabadani, northwest of the capital.
Activists on their Facebook page Syrian Revolution 2011 defiantly called for fresh protests on Friday, the Muslim weekly day of prayers and rallying point for demonstrations.
The rallies will be held under the slogan of "death rather than humiliation," it said. "We are ready to die in the millions as martyrs."
In the YouTube video posted late on Wednesday, the attorney general of the flashpoint rebellious province of Hama, Mohammed Adnan al-Bakkour, announced his resignation.
He said he took the decision after hundreds of jailed peaceful demonstrators were killed by the authorities and buried in mass graves, and 10,000 were arrested arbitrarily.
But the official SANA news agency, which reported on Monday that Bakkour had been kidnapped en route to work with his driver and bodyguard, quoted officials as saying his statement had been made under duress.
It quoted Hama governor Anas Naeem as saying that "Bakkour was forced by his captors to give false information."
Bakkour also cited the deaths of about 320 people under torture at Hama police stations, "the arbitrary arrest of about 10,000 people" and the demolition of homes by the army while occupants were still inside.
The United States denounced what it called the Syrian regime's "abhorrent abuse" of prisoners, following an Amnesty International report of 88 deaths in custody between April 1 and August 15, including 10 teenagers.
In at least 52 of the cases, Amnesty said "there is evidence that torture caused or contributed to the deaths," citing signs of violent beatings, burn marks and cuts.
Assad's regime has defied Western sanctions over its deadly crackdown on dissent, blaming "armed terrorist gangs" for the violence.
"Syria must be allowed to move forward," Clinton said on Thursday. "Those who have joined us in this call must now translate our rhetoric into concrete action to escalate the pressure on Assad and those around him."
This pressure must include "strong new sanctions targeting Syria's energy sector to deny the regime the revenues that fund its campaign of violence."
The European Union is to formally adopt a ban on Syrian oil imports on Friday, but the embargo will not take effect until November 15 for existing contracts after Italy insisted on a delay, according to diplomats in Brussels.
They told AFP that the EU would also expand its list of people targeted by an assets freeze and travel ban.
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