Flicking through the pages of the men's weekly Zoo, past countless breasts, a car feature, and images of particularly vicious flesh wounds, Kat Banyard arrives at the classified ads. There's a whole page for phone sex lines, featuring women bent over or with legs spread, asterisks covering their nipples and genitals, beside captions that describe them as dirty slags, sexy cougars and 60+ grannies. Banyard points to one advertising sex line workers who are "just 18" and a bigger ad, on the facing page, promising "Asian Dolls: find your perfect Oriental escort NOW!". She winces slightly. "I find it staggering that high street retailers sell these magazines. I mean, they've been on their shelves for years, but I still find it staggering that they expect customers and employees to be exposed to this and also that they think it's OK to profit from them."

Banyard, 31, is the founder of UK Feminista, a group that trains and supports grassroots feminist activists; since it began, in 2010, she has become one of the country's most prominent campaigners. She wrote the book The Equality Illusion, led a successful campaign to change the law on lapdancing clubs – enabling local authorities to license them more stringently, as sex establishments rather than just cafes – and called for the outlawing of cosmetic surgery advertising. For someone whose work involves taking on powerful, often angry opposition, she is remarkably self-effacing and soft-spoken.

Towards the end of last month, in conjunction with fellow feminist group Object, UK Feminista began the campaign Lose the Lads' Mags. Their first move was an open letter – now signed by 18 lawyers – published in this newspaper, which warned that lads' mags and "papers featuring pornographic front covers" could face legal action. The lawyers pointed out that any shop is, by definition, a workplace too, and that "displaying these publications in workplaces, and/or requiring staff to handle them in the course of their jobs, may amount to sex discrimination and sexual harassment contrary to the Equality Act 2010".

Banyard compares the magazines to the "girlie calendars" so popular in the 1970s, and says "we take for granted now that it's not acceptable to have those on office walls, because we recognise that that creates a really sexist and degrading environment for women. Lads' mags are the equivalent of girlie calendars, and they are on shelves, in people's workplaces, in everyday spaces across the UK." The campaign therefore calls for high street retailers to immediately withdraw all such publications from their shelves.

The arguments about the potential harm caused by sexist or pornographic publications have always been lively. Banyard believes strongly that lads' mags create "a conducive context for violence against women" and points to reports that have criticised them, including the Sexualisation of Young People Review (2010), commissioned by the Home Office. This stated that they "promote an idea of male sexuality as based on power and aggression, depicting women as sex objects and including articles that feature strategies for manipulating women".

There has been other grist to the mill recently. A 2011 Middlesex University study, for instance, found participants had difficulty differentiating between the comments of convicted rapists and those taken from lads' mags. (One sample magazine line read, "a girl may like anal sex because it makes her feel incredibly naughty and she likes feeling like a dirty slut. If this is the case, you can try all sorts of humiliating acts to help live out her filthy fantasy".) A number of former lads' mag editors have also expressed regret for what they once published. Martin Daubney, who edited Loaded for eight years, wrote last year that "looking back, I think magazines like Loaded did give young men a 'taste' for soft porn that led to deeper and darker desires".

Earlier this month, Banyard and her fellow campaigners began targeting the country's largest retailer, Tesco. In the space of a week, she says, thousands of supporters "emailed, tweeted and Facebooked Tesco and told them to prove their commitments to corporate social responsibility aren't all talk". It seems Tesco might be listening. Three days into the online push, the retailer tweeted that it would talk to publishers about wrapping lads' mag covers, and also agreed to meet the campaigners. That will happen soon, and the activists are planning to attend Tesco's AGM on 28 June too, says Banyard, "to make sure that every single shareholder is aware of how hugely harmful and damaging these magazines are".

The campaign certainly isn't without critics. Their comments range from the offhand and offensive – Banyard says she's been receiving "an awful lot of emails telling me to get back in the kitchen, or that what I really need is a big cock up me" – to much more thoughtful, substantive points. One that has arisen repeatedly is that calling for these magazines to be withdrawn is an assault on free speech. There have been a number of campaigns against lads' mags over the years, but most have called either for an age restriction on sale, or for them to be moved to a higher shelf.

Banyard says they're "not calling for any new laws, policies, or regulations". Instead, they're simply asking retailers "to actively choose to stop selling lads' mags, in the same way that right now they're actively choosing to sell them". Another concern that critics have raised is that, if this campaign is successful, other magazines could be targeted using similar tactics – gay magazines with provocative covers, for instance. Lads' mags aren't the only publications to be considered harmful by some, after all. Fashion magazines have often been accused of pushing unattainable beauty ideals, and some weekly gossip magazines arguably objectify and dehumanise their cover subjects by focusing relentlessly on their weight.

Regarding gay magazines, Banyard says: "The legal parameters around sexual discrimination and sexual harassment are very tightly and specifically drawn. If anyone attempts to perpetuate or further their prejudices through legal channels like that, they would have no truck in front of a judge, and that's the clear opinion of the lawyers supporting this campaign." As for women's magazines, "and the messages they send out about beauty ideals, it's clear they have a deeply damaging effect", she continues, "but we're focusing on lads' mags because of the pivotal role that they play in fuelling a culture and attitudes that underpin violence against women".

The campaign is one of several aiming to improve the representation of women in the British media. In many ways it mirrors the No More Page Three campaign, but where that calls for just one page of Britain's bestselling paper, the Sun, to be changed, the Lose the Lads' Mags campaign, if successful, looks likely to ensure the destruction of these titles. The lads' mag market has been plummeting anyway. In the second half of 2012, Nuts sold an average of 80,186 weekly copies, Zoo 44,068 – a far cry from their mid-noughties heyday, when both averaged more than 250,000.

Banyard says she wouldn't lose sleep if the magazines did close. "I think for too long we've failed to worry about the women and girls who are bearing the brunt of the consequences of how these magazines portray women. What about their rights to be safe, and to participate as equals in society? That's what the board of Tesco should be worrying about. It's high time we started prioritising the safety of women and girls over profit."