Tripoli, Libya (CNN) -- Rebels said Saturday that they were gaining traction in a series of offensives in several parts of Libya controlled by ruler Moammar Gadhafi -- claims that the Libyan government swiftly disputed.

Heavy gunfire was heard in al-Zawiya, located about 33 miles west of Tripoli, where rebels had arrived. Rebels entered the city center and managed to cut off the road to Tripoli, they said.

Talha Jwaili, a rebel fighter, told CNN that rebels control al-Zawiya. Ten rebels died in the fighting, four from Zintan and six from al-Zawiya, he said. More than 20 people were reported injured.

A spokesman in the media office of the Military Council of the Western Mountain Region said that rebels control about 85% of the city and that clashes there are ongoing.

CNN could not independently confirm Saturday who was in control of al-Zawiya, nor other details of the rebels' accounts.

A government spokesman downplayed the fighting, telling reporters in Tripoli that the town is "absolutely under our control."

Al-Zawiya is strategically important because the coastal road through the city is the main supply line from the Tunisian border to areas held by Gadhafi.

Government spokesman Musa Ibrahim said fewer than 100 rebels moved in on al-Zawiya from the southern side of the city, but were stopped. Another group of fewer than 50 rebels was waiting inside the city, but "took action too early," he said, adding that the group is still on the run.

"They are mad. ... This is not an advance. This is what you call a skirmish, what you call a suicide mission," Ibrahim said, noting that al-Zawiya is the country's third-largest city.

While forces loyal to the Libyan leader tried to defend the capital from the west, rebels announced that they had cut off an important military supply route to the south.

Sixty miles to Tripoli's south, rebels said they captured the town of Garyan and pushed the Gadhafi forces to the south, cutting them off from the road to Tripoli. The soldiers left behind heavy artillery and ammunition, rebel field commander Adel al-Zintani told CNN.

"We captured many anti-aircraft artillery vehicles, two full fuel tankers and 106 anti-tank piercing artillery and shells," he said. Seven people were injured in the clashes, he said.

"The rebels have 95% control of Garyan and have surrounded a Gadhafi brigade inside a military camp inside Garyan," said the spokesman in the media office of the Military Council of the Western Mountain Region.

Finally, in the opposition-held port of Misrata, missile attacks by Gadhafi forces ended after rebels captured the nearby town of Tawargha, National Transitional Council spokesman Guma El-Gamaty told CNN. That city was being used by Gadhafi forces to launch missiles indiscriminately into Misrata, he said.

Rebels also took a crucial bridge that links Tawargha to Sirte, Gadhafi's hometown and loyal stronghold, he said.

However, Deputy Foreign Minister Khaled Kaim said government fighters had pushed the rebels away from Tawargha and back to Misrata.

Ibrahim also downplayed the rebel claims in other towns.

"Small groups of armed gangs, 50 here, 50 there, some attacked south of al-Zawiya, some attacked north of Garyan, and Tawargha, but they have very weak influence on the ground," government spokesman Musa Ibrahim told reporters. "The people's armed forces are dealing with them, they do not represent a real threat. Tripoli is safe."

He later noted that Gadhafi's forces are "very strong" and have the support of "thousands upon thousands" of armed volunteers.

"You have to remember we are very powerful," Ibrahim said. "The tens of thousands and tens of thousands of volunteers are armed right now. It doesn't matter whether NATO advances or not, whether rebels advance or not, because we will always be able to fight, in a year's time, in two years, in three years."

Five months into the Libyan war, the rebels have won international support in their effort to oust Gadhafi.

They have been aided by NATO airstrikes that began in March after the U.N. Security Council approved a resolution that ordered civilians be protected.

This week, the government accused NATO of killing 85 civilians, including 33 children, in airstrikes Monday near the embattled city of Zlitan.

NATO has said there is no evidence that the strikes killed civilians, though journalists, including CNN reporters, taken by Gadhafi's government to the site of the strikes, reported seeing bodies of women and children.

It was impossible for CNN to confirm the extent of the casualties and if they were all civilian.

Kaim on Friday criticized the United Nations for what he claimed was an organizational silence over claims that NATO has violated the mandate of the Security Council by killing civilians and conducting a naval blockade.

Ibrahim renewed the government's accusations Saturday, saying NATO is aiding the rebels, whom he described as "armed gangs."

The comments follow a statement a day earlier by a spokesperson for Ban Ki-moon, who 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 urged "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.

Ibrahim repeated Saturday his own calls for an end to the conflict, telling NATO: "Stop this aggression and let's talk peace."

CNN's Salma Abdelaziz, Jomana Karadsheh, Yasmin Amer and Kareem Khadder contributed to this report.