Tripoli, Libya (CNN) -- Rebels pushed Thursday northward on three fronts toward the coastal cities of al-Zawiya, Aziziya and Sorman, with their ultimate goal being Tripoli, rebel field commander Adel Al-Zintani told CNN.
Six rebels died and dozens more were wounded in fighting Wednesday and Thursday, he said.
He predicted that the fighters would reach the coastal road that leads to the capital city within days.
Zawiya is strategically important because the coastal road through the city is the main supply line from the Tunisian border to areas held by Libyan leader Moammar Gadhafi.
Regime officials were not available Thursday evening for comment.
Fighting was continuing around Misrata, where rebel spokesman Mohamed Ibrahim said the main hospital reported four rebels killed and 54 wounded.
Meanwhile, in New York, a spokesperson for Ban Ki-moon said the U.N. secretary-general "is deeply concerned by reports of the unacceptably large number of civilian casualties as a result of the conflict in Libya."
Ban is urging "all Libyan parties" to engage with his special envoy, Abdel-Elah Al-Khatib, "and respond concretely and positively to the ideas presented to them, in order to end the bloodshed in the country," the spokesperson said.
His remarks came three days after allegations that a NATO strike in the village of Majer near Zlitan wound up killing 85 civilians --- 33 children, 32 women and 20 men.
The Tripoli government called it "a massaacre" of civilians; NATO has said its warplanes struck two farms used as a staging point for Gadhafi's forces
"This is a legitimate target. And by striking it, NATO has reduced pro-Gadhafi forces capabilities to threaten and attack civilians," said Col. Roland Lavoie, a spokesman for the NATO operation.
"We do not have evidence of civilian casualties at this stage, although casualties among military personnel, including mercenaries, are very likely due to the nature of the target."
But, on Tuesday, officials from the government in Tripoli escorted international journalists to a mass funeral in Majer, and to the site where the government and residents said the NATO airstrike killed civilians.
Journalists saw the bodies of children and women among the dead and wounded at the site of the attack.
It was impossible for CNN to confirm the extent of the casualties.
A spokesman for the Tripoli government, Musa Ibrahim, called it "a crime beyond imagination."
For weeks, NATO warplanes have been bombing Zlitan on a daily basis. Meanwhile, rebels have been pressing toward Zlitan's eastern gates from the nearby opposition-held port of Misrata.
Speaking to journalists in Majer on Tuesday, Ibrahim accused NATO of bombing Majer to clear the way for rebels to advance on the Gadhafi-controlled city of Zlitan, a few kilometers north.
Maj. Gen. Nick Pope, spokesman for the British Ministry of Defence, said Thursday that British aircraft have been in action in recent days "as part of NATO's mission to protect civilians at risk of attack from Col. Gadhafi's former regime."
The latest operations included British aircraft carrying out precision airstrikes on several sites in and around Zlitan, Pope said in a statement.
The report of civilian deaths stirred anger in Washington.
"If such reports are true, it further highlights the increased risks to civilians that has occurred as a result of military intervention in Libya and is further evidence that continued bombing by NATO ... is an inappropriate tool to protect civilians or force Moammar Gadhafi from power," wrote Rep. Dennis Kucinich in a letter to Ban.
"How can they continue to cause the deaths of innocent civilians under the color of international law and not be held accountable?" the Ohio Democrat continued.
The changing nature of who controls what in Libya was underscored Thursday by events in Washington, where the Libyan Embassy officially reopened under the control of the Transitional National Council.
"This is a message that Gadhafi can no more rule Libya," said Ali Aujali, who was accredited Thursday as head of the Libyan mission.
Aujuli said he talked Thursday with State Department officials about the Obama administration's efforts to help the TNC gain access to some $30 billion in frozen Libyan assets.
The State Department had ordered the embassy closed in March and expelled diplomats loyal to Gadhafi. Aujali had resigned his post as the Gadhafi regime's ambassador to the United States in February and has since represented the opposition in Washington.
The United States on July 15 recognized the rebel movement based in Benghazi as Libya's rightful government.
CNN's Jomana Karadsheh, Elise Labott, Joe Vaccarello, Yasmin Amer and Kareem Khadder contributed to this report.