Nine months ago, Jake Mills texted his girlfriend one final time to tell her he loved her – and then he tried to kill himself. "I genuinely felt that I was a burden to a lot of people's lives," the 25-year-old Liverpool comedian says. "A lot of people say suicide is a selfish act but, in that frame of mind, if you're about to kill yourself, you just don't see anything better."

Although Jake had been visiting a counsellor, he was just telling her what he thought she wanted to hear. "She discharged me and told me that I was healthy and better. But actually I wasn't better, I was just better at lying."

Jake was rescued by his girlfriend and has been confronting his depression ever since. But for all too many men, there is no rescue. Last week, millions were shocked by the suicide of beloved actor Robin Williams. The aftermath has provoked a long-neglected debate about mental health and suicide.

A cursory look at the statistics in Britain suggests it is dearly needed. Suicide is the biggest killer of men between 20 and 49, eclipsing road accidents, cancer and coronary heart disease. It is also predominantly a male disorder. Of the 5,981 suicides in 2012, an astonishing 4,590 (76%) were men. And yet while Britain has high-profile campaigns on, say, testicular cancer or driving safely, the biggest killer of men under 50 is not getting the attention it deserves.

Jane Powell is the founder and director of Calm, the Campaign Against Living Miserably, which specifically deals with male suicide. "If you're a mum, a dad, a loved one, you want to worry about the biggest threat," she says. "And yet we worry about assault levels, rather than the real killer – suicide." She makes a provocative case: that while breast cancer does kill men, we rightly focus on it as a female disease. In the same way, suicide prevention has to focus on men. "We need to name the issue," she says.

Why are so many more men killing themselves than women? "Is it biologically set in stone that men take their own lives – or is it cultural?" Powell asks. "If you look at how the suicide rates have changed, how they go up and down, you can see that it's cultural – it's about what we expect." And this is what is so troubling about male suicide. Women are actually more likely to suffer from depression, but more likely to seek help when they encounter trouble. The uncomfortable truth is that stereotypical forms of masculinity – stiff upper lips, "laddishness" – are killing men.

Ant Meads, a 35-year-old based in Coventry, tried to end his life nearly two years ago. Growing up, he had undiagnosed obsessive compulsive disorder (OCD). "I was 18, working in a concrete yard, surrounded by big burly men doing manual labour, and I was a stick insect," he recalls.

Whenever his hands were dirty, he felt nauseous, so he wore gloves all the time. "I got bullied every single day when I was there, I was the 'little special princess who has to have his special gloves'. It was the first time I realised that I was not living up to the ideal of what a man should be."

His OCD would get worse, until he was too anxious to leave the house. "I was failing myself, my family, society, because I couldn't do what every man was supposed to be able to do." He was left with a sense of worthlessness, of letting people down.

When he first told his father that he was depressed, he told him to "get over it". It wasn't just relatives: doctors told him to "get on with it" as well. When a doctor finally referred him to psychiatric care, Meads faced a six-week wait before finally being seen by a community liaison officer, who disputed whether he had OCD because – unlike other patients – his hands were not chapped. "It's this horrible idea of what a man is supposed to be," he explains. "It's a general feeling, evident in the fact that so many men commit suicide, because they're not living up to this mythical idea."

This sense that men should not speak about their feelings is not always overt; nor does it necessarily manifest itself as bullying. Fabio Zucchelli, 29, has had depression since his early teenage years. "I noticed that I was low for very long periods, and it developed into what people term clinical depression," he says. "It certainly held me back in many ways, up to my early to mid 20s ... There have been long periods when I've felt not able to work at all."

Zucchelli says he didn't suffer from "self-stigmatising issues", and has been able to talk about his feelings with professionals. "The main issue I've had with feeling able to talk about mental health difficulties is with male friends, who just find it really uncomfortable. I haven't had anyone defriend me because of it, just a lot of discomfort."

When the 35-year-old Labour MP John Woodcock announced last December that he was depressed, he was confronting a double stigma: not just as a man talking about his mental health, but as a politician discussing a personal issue that is all too often portrayed as a weakness.

"I've been really struck by the number of men who have come up to me – often in my constituency – like ex-shipyard workers who have struggled for 10 years, who have been keeping it quiet," he tells me. "We do operate in a culture where men, by and large, talk about their feelings less. They're self-conscious about talking about weakness, there's this male sense of 'shrug and get on with stuff'."

This type of male identity is cemented at a very young age. According to research by the LGB charity Stonewall, 98% of gay pupils and 95% of teachers hear "that's so gay" or "you're so gay" at school; nearly as many hear "dyke" or "poof" thrown around as insults. "It's so much wider than gay or bisexual men," says Stonewall's spokesman Richard Lane. "Men hear 'man up' and 'stop being such a poof'. It's a real barrier in talking about mental health issues."

Rather than being entirely about anti-gay hatred, there is an element of "gender policing", of abuse directed at men who do not conform to a stereotype of masculinity.

"Asking for help is seen as an affront to masculinity," says the writer Laurie Penny, who has extensively researched mental health issues and written about her own experiences. "This is deeply, deeply troubling, because it means when you're taking that first step when you're suffering a mental health difficulty, reaching out for help is made doubly hard. The rules of masculinity prevent you from asking for help or talking about feelings."

According to Penny, depression is often accompanied by a sense of shame, of not deserving help, "and when messed-up gender roles are thrown into the mix, it's going to become even more troubling". She has no doubt that gender policing "ruins lives across the board".

Mind is one of Britain's main mental health charities; according to its research, just 23% of men would see a GP if they felt low for more than two weeks, compared with 33% of women. "One of the more common ways men deal with it is self-medicating with alcohol and drugs," says the Mind spokeswoman Beth Murphy. "They start going to the pub, block feelings, hide feelings, drink, then do it more, and it becomes a cycle. The drugs and alcohol can end up as big a problem as the mental distress in the first place." Indeed, research has suggested that men are twice as likely as women to develop alcoholism.

In the late 1990s, it was men in their 20s who were most at risk from suicide; today it is men in their 40s. As Murphy points out, it's the same cohort, and is evidence of "scarring": of being unemployed at a young age, and suffering from long-term consequences, including higher rates of unemployment and lower wages in later life, as well as mental distress.

According to research by Samaritans, those in the poorest socioeconomic circumstances are 10 times more likely to kill themselves than those in the most affluent. Both men and women experience poverty, of course – but it is men who are more likely to kill themselves if they are poor.

And the help simply is not there for men, even if they seek it. When Ant Meads finally saw the doctor who instantly recognised his OCD and began a referral for specialist care, he faced a nine-month wait. "Imagine you're suicidal, you need to see a psychiatrist, and you're told the current waiting list is nine months. How do people cope?" Meads is adamant that he would not be alive had his employer not referred him to private healthcare. He believes there needs to be far more government investment and a national advertising blitz about men and mental health.

But the winds are blowing against Meads's calls: mental health trusts are making cuts amounting to 20% more than those made by other hospitals; mental health services have cut beds by nearly 10% in the past three years, and mental health organisations have warned that cuts to such services are risking people's lives.

"I'm really concerned about it," says Woodcock. "We've now made the commitment that mental health should have parity with physical conditions in the NHS, but we're not close to delivering that parity."

Unless this changes, he suggests, "we'll fail many people who will suffer mental health conditions, or even suicide, when they can be helped."

Challenging unreconstructed masculinity is surely a priority, too. The organisation Calm has launched an initiative called "#mandictionary", encouraging men to take on "archaic male stereotypes" and "define themselves on their own terms". Men speaking out – as they have done in this article – helps, too, encouraging others to come forward. "From a personal point of view, I'd never admit to anyone I was depressed, I didn't even want to be on antidepressants because of the stigma attached," Jake Mills explains.

"I thought I'd be addicted to them and weak." Now he uses his comedy to raise the issue. "The best decision I made in my life was announcing it, going on Twitter. I've had enough, I'm not hiding from it any more."

Speaking out and challenging the stigma of mental health is certainly courageous. And doing so may just help to save the lives of other men who are suffering in agonising, lonely silence.