That two of the racist thugs who murdered Stephen Lawrence have been locked up is, finally, some justice. But this is no moment of catharsis; nor can we say this is the long-awaited righting of an extreme injustice which we can put behind us. That's certainly true with the Lawrence family: they have still been robbed of their son, and will grieve for the rest of their lives. Some of the other racist gangsters who murdered him nearly two decades ago remain at large.

But it's also true about the fight against racism. Doreen Lawrence has been an inspiration throughout her family's 18-year ordeal. And she has a message for all of us that should be a rallying cry in the aftermath of this verdict: "The fact is that racism and racist attacks are still happening in this country, and the police should not use my son's name to say that we can move on." With nearly 40,000 race hate crimes in 2010 (with many others undoubtedly unreported), it is no time for complacency in the struggle against all forms of prejudice and bigotry.

That is not to gloss over what has already been achieved. On the contrary, we should draw strength from it. Just over 50 years ago, a Gallup poll found that 71 per cent of Britons opposed interracial marriage. The number of people who still hold this view is so small that pollsters have stopped recording the figure. Indeed, Britain has the highest levels of mixed-race marriages in Europe. Only 3 per cent of people now admit to being "very racially prejudiced": undoubtedly many others are not being honest, but the fact that the figure has fallen is testament to the degree to which overt racism has become unacceptable.

We should also not fall into the trap of portraying racist gangsters like Gary Dobson and David Norris as somehow emblematic of the bigotry of the so-called "white working class". Of course, racism remains a problem within working-class communities as it does at every level of society. But inner-city areas are far more mixed than many leafy middle-class suburbs. Some 35 per cent of London supermarket workers hail from an ethnic background; the figure is 10 times lower among partners of Britain's top 100 legal firms. In London Boroughs such as Hackney, Tower Hamlets and Newham, working-class people of all ethnic backgrounds work, socialise and sleep together. Around half of British-born black men are in relationships with white partners, and it is in these sorts of communities where such mixing is taking place most.

But race and racism continue to loom large over British society. It is a point that was made to me forcefully in the aftermath of the August riots. I spoke to a number of young black men about their experiences with the police. Like me, they had never been charged with a crime. But while I have never been stopped and searched by the police in my life, it was an experience many of them had had to endure since they were as young as 12. Sometimes, the officer stopping them was respectful, even almost apologetic; but at other times they came across as aggressive or intimidating. For some of the young men I spoke to, the police acted as though they were "the biggest gang around here". It is a shocking statistic, but black people are 26 times more likely than whites to be stopped and searched by the police under Section 60 of the Criminal Justice and Pubic Order Act in England and Wales.

Some parts of the media retain their pernicious role of fanning prejudice. Because of its relentless pursuit of Stephen Lawrence's murderers, the Daily Mail is receiving much kudos, including from its normally diehard left-wing critics. But let's not forget that newspaper's role in promoting dangerous myths about immigrants and ethnic minorities. Take these Daily Mail headlines: "Maternity units turn away British mums as immigrants' baby boom costs NHS £350m"; and, "Want to see a GP? Gypsies come first as NHS tells doctors that travellers must be seen at once." This is the kind of reporting that perpetuates the dangerous myth of white Britons being undermined by ethnic minorities – and thus helps inflame divisions within our society.

Some forms of prejudice have actually become worse since Stephen Lawrence was murdered. I'm not one to normally agree with the Conservative Baroness Sayeeda Warsi but she was right to say that Islamophobia now passes the "dinner-table test". It is a bigotry even indulged by some progressives. The 19th-century German socialist August Bebel once described anti-Semitism as the "socialism of fools" because of its rhetoric about Jewish financiers; Islamophobia today could be described as the "secularism of fools". A study at the end of 2007 revealed that 91 per cent of articles about Muslims in one selected week of coverage were negative; I doubt things have improved since. For the British National Party – as is the case for the far-right across Europe – Muslims are now the main target of choice.

That's why it's so important that we take heed of Doreen Lawrence. Racism and prejudice retain their ugly presence at every level of society. The fight against it has a long way to go.

Owen Jones is the author of 'Chavs: The Demonization of the Working Class'