The murder of the teenager Shafilea Ahmed is likely to stand out in British history as a particularly gruesome example of what we now refer to as "honour killings". Shafilea had warned that her parents were trying to marry her off to Pakistan; others knew she had sustained injuries from beatings by her parents; she had even tried to commit suicide in Pakistan. But right until the end, her own life was fated to be out of her control - she went missing in 2003 and her dismembered body was found a year later.

Shafilea's case wasn't a one-off. It took eight years for the murderers of the Sikh woman Surjit Athwal - her own husband and mother-in-law - to be brought to justice in 2007. It took ten years for Mehmet Goren to be jailed, in 2009, for murdering his daughter Tulay, because she fell in love with an older man of a different Muslim sect. Last year Gurmeet Singh Ubhi was found guilty of murdering his 24 year old daughter Amrit because the Sikh girl was dating a white man. There are others too - Heshu Yones, Banaz Mahmod, Nuziat Khan - the list of women murdered in the UK over their "honour" is depressingly long.

Any decent person would be angry and sad when presented with these names. But it isn't enough to be sad: we have become complicit in this epidemic of abuse and violence by not doing more to challenge it. This should be a left-wing cause célèbre but instead there is an embarrassed silence. Left-wing activists robustly challenge racism and homophobia - so why isn't more being done to stand up to this social evil?

Perhaps it's because forced marriages and honour killings are fundamentally about a lack of cultural integration. They almost always happen when parents want to stop their daughters from mixing with British culture and life, instead of abiding by outdated cultural traditions. To these parents, mixing with other cultures or ethnicities has become synonymous with losing the "family honour".

In the past, most left-wing intellectuals and activists avoided the issue of cultural integration for good reason: it was usually a code-word for assimilation and conjured up images of whites telling others their way of life was inferior. But caught between attacks from the right on multiculturalism, and worries that speaking out would look racist, we have become paralysed. Meanwhile, these problems are set to get worse as more second and third generation children of immigrants come of age.

The Iranian and Kurdish Women's Rights Organisation (Ikwro) found last year that 39 out of 52 police forces across the UK had recorded at least 2,823 "honour" attacks over 2010. Some forces showed a jump of nearly 50 per cent in such cases from 2009. Horrifying stories of friends subjected to subjected to verbal abuse by parents for straying past cultural boundaries are far too common. One acquaintance of mine, under pressure to marry before she was ready, was repeatedly told she was being "too fussy" and would end up "left behind on the shelf". Her self-confidence was broken down with every barbed comment until she eventually gave in. Emotional blackmail is one thing, but the Crown Prosection Service estimates that on average one girl is murdered every month in Britain over "family honour" - while the number of forced marriages could be as high as 10,000 a year.

Jagdeesh Athwal, who fought for nearly a decade to get justice for his sister Surjit, told the Independent that Asian community leaders and religious groups "remain deafeningly silent when these killings happen". The author Sufiya Ahmed, whose book, Secrets of the Henna Girl, is about forced marriages, admits there is "a lot of denial" about the problem. "Aside from not wanting to acknowledge that it happens, there are even those who argue that the subject shouldn't even be talked about for fear of branding all Asians with the same brush."

There is a patronising attitude at work here, too: well-meaning liberals have been known to assume forced marriages and such abuse are an inherent part of Asian culture and therefore must be left alone. A few months ago, activist Jasvinder Sanghera's charity Karma Nirvana convinced a teacher in East London to put up posters at the school that said: "Forced Marriage is Abuse Not Cultural". But the posters were taken down and the charity was told the head was concerned they would upset Muslim parents.

A reluctance to offend will only encourage more silence. If self-appointed community leaders won't speak out, then Labour politicians and left-wing activists have a duty to. I'm not calling for a sneering attitude that says, "if you want to be British, you have to integrate into our way of life". Instead, we need an inclusive narrative about integration which is clear that we, as Britons, are equal regardless of religion, ethnicity or background. It has to appeal to both white and non-white Britons. It has to be about what it means to live in this country collectively and what we do tolerate as a society and what we don't. It would give us more confidence to stop the systematic abuse of thousands of women in Britain. Otherwise, we stay complicit in their fate.