Tens of thousands of children have been sexually abused in Zimbabwe in a growing epidemic that has shocked human rights activists.

A single clinic in the capital, Harare, says it has treated nearly 30,000 girls and boys who were abused in the past four years ‑ an average of 20 per day. Experts believe that the country's economic collapse under Robert Mugabe has led to widespread family breakdown and left many children vulnerable.

Dr Robert-Grey Choto, a paediatrician and co-founder of the Family Support Trust Clinic, said the increase was alarming. "In the last four years we have seen over 29,000 cases, and in the last 10 years we have more than 70,000 at this clinic alone," he told the BBC's Network Africa programme. "It's a tip of the iceberg ‑ the problem is enormous. We need drugs and any assistance we can get."

A 12-year-old patient at the clinic, part of the main referral hospital in Harare, told the BBC he had been gang-raped in a township last month. "Four men waylaid me on my way from school," he said. "I was taken to a shop where they showed me pornographic material."

The boy said he was then drugged and sodomised for more than a week. His father added: "This is unbearable. All I want is justice for now."

Other organisations dedicated to helping victims are on the back foot because of Zimbabwe's tense political climate. Betty Makoni, founder of the Girl Child Network (GCN), which has rescued more than 35,000 girls from sex abuse, was forced into exile last year because of threats against her.

Speaking from London, she said the real number of victims was likely to be double that recorded by the Family Support Trust Clinic. The GCN says 10 girls report rape every day in Zimbabwe and a further 10 victims probably remain silent. The youngest known victim was a baby of one day; the oldest was a woman aged 93.

Makoni told the Guardian: "We have children forced to marry under the age of 13. We have children who were held hostage and raped in militia camps during the political violence who are now giving birth to their own children. We still have children being raped because of the myth that if a man with HIV has sex with a virgin he will be cured of his virus."

She said men were able to perpetrate the crime with impunity because of 4,000 known rape cases per year, only 500 resulted in a prosecution. The GCN's research indicates that on average a man can rape 250 children before his crimes become public knowledge.

"The justice system has collapsed in Zimbabwe. A syndicate of men uses its economic and political muscle to escape justice. We also have 10,000 boys going to train as youth militia; they become vicious and make girls succumb to sex through fear."

The economic meltdown, political violence and starvation in Zimbabwe over the past decade have driven numerous people abroad, with 3 million fleeing to South Africa alone. Often they leave their children in the care of extended family or friends and try to send money home.

Many more children have been orphaned by HIV/Aids or other diseases in a country where the average life expectancy has plummeted to 37 for men and 34 for women, among the lowest in the world.

Chipo Mukome, a counsellor at the Family Support Trust Clinic, told the BBC: "Due to the economic situation where we have seen a lot of parents going to neighbouring countries, like South Africa, in search of greener pastures, they are leaving their children to the care of others ‑ uncles and aunts for example. These people, in the end, are abusing these children."

Zimbabwe's fragile unity government has limited capacity to intervene after years of neglect of welfare state structures. The priority in recent months has been the reopening and maintenance of crumbling schools that were once the envy of Africa.

David Coltart, the education minister, said: "I suspect that a third of households in Zimbabwe have been broken up as a result of the economic chaos. But the social welfare department has all but collapsed. There are hardly any social workers left."

Coltart, a member of prime minister Morgan Tsvangirai's Movement for Democratic Change, said the child sex abuse statistics were indicative of a wider epidemic. "In the last few decades we allowed a culture of violence to pervade our society," he said. "It's compounded by the fact that those responsible are generally immune from prosecution. The breakdown of the rule of law means this culture is all-pervasive. It is not just intra-political parties. It spreads to domestic violence and the abuse of children."

Last month Coltart launched a campaign, Learn Without Fear, aimed at ensuring schools are safe places for children. It noted that while teachers have been responsible for abusing girls in schools, there has been a developing trend in which girls are abused by senior boys, with some cases going unreported.