Western-backed African Union troops in Somalia gang-raped women and girls as young as 12 and traded food aid for sex, Human Rights Watch has said.

An investigation uncovered evidence of sexual exploitation of women seeking medicine for sick babies at what they assumed was the safety of AU military bases.

The 22,000-strong AU force in Somalia, known as Amisom, with soldiers drawn from six countries, has been fighting alongside Somali government troops against the Islamist militant group al-Shabaab since 2007. Its donors include the UK, US, European Union and United Nations.

On Monday, Amisom claimed that alleged rapes were isolated incidents and described the report as "unbalanced and unfair".

Human Rights Watch's 71-page report, The Power These Men Have Over Us, documents cases involving troops from Burundi and Uganda. Amisom also draws military personnel from Kenya, Ethiopia, Djibouti and Sierra Leone.

The vulnerable women largely came from camps in the capital, Mogadishu, having fled rural Somalia during a famine in 2011. The youngest girl interviewed was 12 and said she had been raped by a Ugandan soldier.

The mother of one girl who was allegedly raped told the group that she was deeply traumatised. "People laugh at her whenever she comes out," the mother said. "They say: 'An infidel raped her.' How can you feel if your daughter asks you: 'Mother, I better die to hide my shameful face from the people?'"

The report, based on testimonies of 21 women and girls, says: "Some of the women who were raped said that the soldiers gave them food or money afterwards in an apparent attempt to frame the assault as transactional sex."

In May 2013, for example, Kassa D was introduced to a Somali interpreter at Amisom's base camp. "I was worried," she said. "I wanted to run but I knew that the same thing that brought me here would get me through this – my hunger. I had made a choice and I couldn't turn back now." After she had sex with a Ugandan soldier, the interpreter paid her $10 (£6).

The AU soldiers, "relying on Somali intermediaries, have used a range of tactics, including humanitarian aid, to coerce vulnerable women and girls into sexual activity," the report continues. "They have also raped or otherwise sexually assaulted, women who were seeking medical assistance or water at Amisom bases."

Women reported contracting sexually transmitted infections, mainly gonorrhoea, after the assaults, and said soldiers refused to wear condoms. "Several also described being slapped and beaten by the soldiers with whom they had sex," the report adds.

In only two cases had women who spoke to the watchdog filed police complaints. Most had not because they "feared stigma, reprisals from family, police and al-Shabaab", the report says.

Somalia is struggling to recover from two decades of conflict and anarchy. Last month, Amisom troops launched an offensive aimed at seizing key ports and cutting off an important source of revenue for al-Shabaab, which has sworn allegiance to al-Qaida. Last week al-Shabaab's leader, Ahmed Abdi Godane, was killed by a US air strike.

On Monday, at least 12 people were killed and dozens injured in a suicide car bomb attack on Amison troops. Al-Shabaab claimed responsibility for the ambush.

Human Rights Watch said it conducted the research for its report on Amisom in Somalia, Uganda and Burundi. It also interviewed more than 30 witnesses, foreign observers, military personnel and officials from troop-contributing countries.

It said: "The findings raise serious concerns about abuses by Amisom soldiers against Somali women and girls that suggest a much larger problem." It called on the AU and Amisom to foster a culture of zero tolerance towards illegal activities on their bases and set up or strengthen disciplinary units and an independent investigative body.

Daniel Bekele, Africa head at Human Rights Watch, said: "The AU military and political leadership needs to do more to prevent, identify, and punish sexual abuse by their troops. As another food crisis looms in Mogadishu's displacement camps, women and girls are once again desperate for food and medicine. They should not have to sell their bodies for their families to survive."

Amisom's commander, Gen Silas Ntigurirwa from Burundi, told Agence France-Presse that the report documented "allegations of isolated cases of rape", and insisted that his soldiers were given strict orders against raping and looting.

Amisom spokesman Eloi Yao said the "allegations will be properly investigated and measures taken".