As local elections approach, the danger of overconfidence becomes a very real one for Labour. A year ago the Tories were sure they would snuff Labour out as an electoral force; instead Corbynism deprived Theresa May of a majority, ending a neoliberal consensus that had prevailed for a generation. The result left the party membership and much of its support base exuberant. Yet the Labour party, nationally and at London level, is guilty of failing to manage expectations. Despite recent government chaos over the Windrush scandal, Labour has spent the last few weeks in a defensive posture against a media onslaught that has undoubtedly had an impact.

Yes, polls suggest Labour is on course to win the best results of any party in the capital for more than 40 years. But George Osborne’s London Evening Standard and No 10’s spin doctors have hyped up the prospect of Conservative Armageddon – hoping to frighten Tory voters to polling stations, suppress the Labour vote through complacency, and make any Labour gains seem insufficient. Unless the party’s grassroots turn out to vote, historic prizes will remain out of reach.

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Labour’s electoral goals would, in a not too distant age, have been seen as otherworldly, and the party’s current challenges nice to have. Westminster, Barnet and Kensington and Chelsea are all Tory citadels in the capital that have never surrendered to the wooing of Labour, and the party has never won a majority in Barnet, not even in the age of the poll tax or a national Conservative meltdown. Wandsworth, in south London, was last in the red column when James Callaghan was prime minister four decades ago: Labour has never amassed the number of wards it needs to win there in one go. One ward Labour needs to gain in Wandsworth to win the council has an average property value of £1.3 million. Labour is seeking to win wards in Westminster and Kensington and Chelsea, which are home to Buckingham Palace and Harrods. Outside London, only a third of some councils’ seats are up for grabs, making Portsmouth impossible to win, and Trafford, in Manchester – last held by Labour 15 years ago – extremely challenging.

Labour’s big asset, of course, is a mass membership: I’ve spent the last few weeks with Momentum, mobilising thousands of activists as part of a national campaign called Unseat. Many are young people who are campaigning for the first time – such as the 19-year-old daughter of two Norfolk Ukip voters, or 17-year-old Frank, who travels to campaign days from Lewes in East Sussex. This politicisation of young people isn’t covered enough: the idea of teenagers taking trains and buses across the country en masse to canvass in local elections would be far-fetched in my own younger days. This campaigning is particularly critical for Labour. Those most likely to vote are older, white, more affluent people, who strongly lean towards the Tories.

Facebook Twitter Pinterest Labour MPs Rosena Allin-Khan (C) and Harriet Harman with candidates and activists in Wandsworth on 29 April. Photograph: Wiktor Szymanowicz / Barcroft Images

A perennial problem is that those suffering most from a broken social order are the most disillusioned about the entire political process. The reach of Corbynism still has limits. In a Westminster tower block, for instance, I meet a man who lives with three others in a one-bedroom flat: the Tory borough won’t rehouse him because there aren’t enough council houses. In Trafford, a tired-looking woman in her early 30s opens the door, cradling her baby. She fled her husband who beat her, and she and her two children now live with her parents in a cramped two-bedroom house, causing huge family tensions; again, the council has no stock to rehouse her.

Both are disillusioned with politics and had no intention of voting. It takes a prolonged conversation to convince them otherwise. There are younger voters who may spontaneously break out into chants of “Oh, Jeremy Corbyn!” but remain distinctly unenthused by local elections. Some have got in touch to ask me what local elections are, which is unsurprising in a country that does so little to educate its citizens about politics.

The campaigning is critical for another reason. Labour began the last general election campaign with catastrophic polling. Critical to reversing the party and leader’s poor popularity ratings were the broadcasting impartiality rules, which kicked in after the election was announced, granting Corbyn unfiltered access to the public. The dramatic increase in political interest gave an opening for a social media campaign that reached millions of voters. Combined with a transformative manifesto, the shift in public attitudes in a six-week campaign was unprecedented.

Without a determined push against complacency the great Tory prizes will remain out of reach

In these elections, Labour has faced a relentless media barrage, with very few sympathisers in the press to challenge it: the party’s enemies are defining it again. Its policies – such as free bus travel for the under-25s, or linking housing affordability to people’s incomes – were covered in the press, but received no broadcast coverage. National media have shown extremely limited interest in the elections, leaving it off the radar for many. Doorstep campaigning therefore takes on a particularly critical function, both bypassing a hostile media and raising awareness that there are even elections on.

But there are profound challenges to Momentum’s mass canvasses. Some local parties in Tory-held areas, which until recently were only husks, are unaccustomed to mass canvasses, meaning such campaigning is sometimes chaotic. “Data is good in terms of Labour promises but canvass days are tiny, as are members signed up to door-knock on election day,” says one councillor in Wandsworth. “Just nowhere near the same numbers as the general election. I think the membership are on a high, and think things are all good, but they don’t realise how important getting out the vote is for locals, due to turnout.”

I’m told that Tory core voters are fired up, fearing a clean sweep by the red menace, while many Labour supporters are relatively uninterested in local elections. In Wandsworth the local party believes that it could have enough promises to win the council – but fears it doesn’t have enough activists to turn out the vote on election day. Such warnings sometimes require a healthy scepticism: on the evening of the general election, one senior Tooting Labour figure texted me to warn that turnout was rocketing in Tory areas and suppressed in Labour’s. A few hours later the Labour majority surged from less than 3,000 to 15,458. But local Labour fears about turnout on the ground are genuine.

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Barnet, in north London, is home to Britain’s biggest Jewish population, many of whom feel angry about, and fearful of, the left’s antisemitic fringe. At a general election, Labour could get voters’ attention with promises to end austerity, scrap tuition fees, introduce a living wage and bring utilities under public ownership. Commitments in local elections are inevitably more modest, which makes arousing enthusiasm more challenging. Labour is making pledges to use local powers, for instance, to challenge the housing crisis, but the devastating government-imposed cuts limit the party’s ability to match the radicalism of its national policy commitments. In addition, Brexit is an understandably emotive subject among citizens of the European Union who are entitled to vote.

Media smears, too, can land: one middle-aged voter in Trafford, previously Labour, yelled at me that Jeremy Corbyn was a Russian spy who would make the tsar ruler of the country, and that he needed “shooting”. It is a reminder that the media is fuelling dangerous radicalisation.Corbyn’s opponents – the Tories, the press, and within the Labour party – are desperate to turn the clock back on the last general election result – to restore a narrative that the left is a doomed, delusional cult. They may be attempting to set the party ludicrously high standards in the local elections to justify another round of attacks. Labour will undoubtedly do well. But without a determined push against complacency, and to mobilise the party’s base, the great Tory prizes will remain out of reach.

• Owen Jones is a Guardian columnist