(CNN) Kurdish families huddle on blankets in a dimly lit cave. Others hide in the rubble of a bombed-out building, gathered around a campfire. Those that have basements seek shelter there.

This is life in Afrin.

Turkey's cross-border military offensive against US-backed Kurdish militias has driven civilians living in the northwestern Syrian enclave underground. An estimated 16,000 people have been displaced by the fighting, which has been punctuated by relentless airstrikes and shelling, according to the United Nations

Footage obtained exclusively by CNN captures the city's deserted streets -- littered with crumpled cars, debris and gaping holes where shops once stood.

"We don't know where to go," says Mohammed Khaled, 10, speaking in Kurdish.

"The airplane has been dropping bombs for five days now. They are dropping missiles and bombs. My dad says not to go out because of the war planes and so we are sleeping in the huts. All our homes are destroyed."

Khaled stands outside one of the structures where civilians are seeking shelter. Children mill around behind him, restless. According to the United Nations Children's Fund (UNICEF) , families have been prevented from leaving the area by local authorities. The humanitarian organization has had to suspend child protection services there amid the violence.

Um Muhammed, Khaled's mother, asks why Turkish President Recep Tayyip Erdogan is bombing her people.

"What did we do to him?" she says in Arabic.

"We lost our homes our children, nothing is left ... why would this happen to us? Is not this a shame that children have to live like that? We are human beings, are we not? Why are they doing this to us?"

"This is a massacre," she adds, breaking down in tears. "Please convey our message we are pleading with the international community to stop the killing of the civilians stop the airstrikes and the war against us."

Afrin has borne the brunt of Turkey's attacks since January 20, when Ankara launched Operation Olive Branch to remove Kurdistan Workers' Party (PKK), Kurdish Democratic Union Party (PYD) and ISIS militants from the area along its border.

The military operations have predominately targeted Kurdish fighters of the People's Protection Units, or YPG. The militia, which has been critical to the US-backed Syrian Democratic Forces (SDF) in the fight against ISIS, is viewed by Ankara as a terrorist organization.

A plume of smoke billows over Afrin as bombardment continues.

Turkey sees the quest by the Kurds -- who are spread out in Turkey, Syria, Iran and Iraq -- to establish an independent homeland as an existential threat to its territorial integrity. And Turkey has long warned that it will not tolerate YPG control of much of its border with Syria.

Hevi Mustafa, the co-president of the executive council of Afrin, praises the resolve and confidence of the forces fighting the onslaught.

Mustafa calls the Turkish attacks "barbaric" and says she hopes the international community will hold the Turkish government accountable for the violence.

"Our soldiers are fighting fiercely. They took on the defense of Afrin themselves," she says. "We expected this attack on our areas because we are part of a democratic project and wanting to end the Syrian crisis in this project. Of course, the Turkish government, they don't want to end the crisis in Syria."

A child rests after being treated in Avrin Hospital in Afrin.

On Wednesday, the SDF said an "invasion army" of Turkish fighters and terrorists with Al-Nusra, al Qaeda's former affiliate in Syria, pounded Afrin with Katyusha rockets, targeting Afrin's largely civilian Ashrafia neighborhood and wounded people were taken to hospitals. The SDF said civilians, from children to the elderly, were injured.

"Right now we are overwhelmed with injured and killed civilians," says Dr. Jawan Muhammed, general manager of the hospital in Afrin.

"Our hospital is unable to cope, Our surgery rooms are overwhelmed. We conduct 18 surgeries a day. We are using up all our medical supplies because of the overwhelming number of casualties as a result of indiscriminate airstrikes and artillery fire."

Turkish forces on Wednesday reported more deaths in the region, citing rocket fire and shelling, but blamed them on "terrorists" linked to the PKK.

A wounded Kurdish man talks on the phone at the hospital.

Turkey views the YPG as indistinguishable from the PKK and says those entities have been hiding out in Afrin since Syrian President Bashar al-Assad's regime left in 2012.

According to the Turkish General Staff, the operation is being carried out under the framework of Turkey's rights based on international law. The military also says avoiding civilian casualties is of the "utmost importance."

"Only the terrorists and their shelters, barracks, weapons, tools and equipment are targeted, and all kinds of attention and sensitivity are shown to avoid damage to civilian/innocent people and to the environment," the Turkish Armed Forces said in a statement.

The people hiding in Afrin's caves have a different view.

One woman asks, "What are the planes wanting from us? What are they bombing us for? What do they want from us and what do they want from the little children?"

A young girl named Yasmin went into hiding with her mother and her brothers after her father was killed.

Fatima Muhammed (right), is sheltering in a cave with about a dozen others.

"It is really dark here. We are so scared because it is really noisy. They are conducting airstrikes. What did we do to them? We are just kids. Why is this our fault?" she asks.

Fatima Muhammed, bundled in a red sweater, her hair wrapped in a purple scarf, is among about a dozen people hiding out in the same cave.

"We ended up in the streets and in the caves ... we can't go back to our homes, they are all destroyed," she says. "What kind of a country strikes civilians in this manner?"

This story has been updated to correct a misattributed quote.