Joe Green, 19, looks a little uncomfortable. He is perched on the edge of an armchair in a freshly decorated house at a secret location in Manchester. He has agreed to talk if I don't give his real name or reveal too many details about him or where he lives.

Four years ago, Green came out as gay to his family and was rejected. His relationship with them has been strained ever since and he can never live at home again, he says. The four-bedroom house he has moved into was opened this summer by the charity the Albert Kennedy Trust (AKT), for lesbian, gay, bisexual and transgender (LGBT) young people.

Setting up the house, and another in London, was a long-held dream of the 24-year-old charity, which was founded in memory of a 16-year-old gay man who, having run away from a children's home, was found dead at the bottom of a multi-storey car park in Manchester. He had been chased by "queer bashers". Every year it is contacted by hundreds of desperate LGBT young people who have nowhere to go.

Green and his housemates, all in their late teens and early 20s, three male and one female, will stay for about six months at the Manchester house while they work out more long-term accommodation with the help of the charity.

"Before this, I was living with an AKT carer in supported lodgings for two years," Green tells me. "It was really good, but this is better for my independence."

Lucy Bowyer, who manages AKT's housing project, Purple Door, sits in while I talk to Green. "Some of the young people have been threatened with physical abuse by their families or are at risk of forced marriage; some are sleeping in stairwells because they can't go home," she says.

The Manchester house – procured from a private landlord in partnership with charity Threshold and New Charter Housing Trust Group – is spacious and tastefully kitted out thanks to volunteer interior designers. A large print of Audrey Hepburn sits on the wall opposite Green, and a cabinet next to him has a small collection of DVDs and games.

"My relationship with them has improved," he says, when asked about his family. "But, they are very religious and I know there's no way I'd be able to live at home ever again."

Green and his housemates are regularly visited by AKT staff, who help with job and life skills and support them to get work or to further their education, as part of the housing project. "I like it here. It's good to be sharing with people your age and, it's more comfortable with everyone else being gay too," Green says.

In the London house, which also opened this summer, through a partnership with housing trust Circle 33, young people aged 18 to25 generally stay for around three weeks as emergency cases.

Bowyer says they sometimes arrive with nothing because they have had to flee in the middle of the night. "Often the young people we work with have come to the end of the line of friends they can stay with; some have been sleeping on buses, not knowing where to go. Some have been in supported accommodation where they've been bullied," she says.

Each resident receives a welcome pack provided by a network of LGBT employees who work for the Co-operative Group. It includes stationery, toiletries and a teddy bear, "in case they get lonely", Bowyer explains.

Hanif Asim, also 19, has been in the London house for nearly two months. He is struggling to find somewhere else to live – a frequent problem in the capital, with high rents and scarce accommodation. AKT can sometimes contribute to deposits, but has a fairly limited pot and it can take time for young people to access affordable credit. Some can move on to stay with the AKT carers, but again, that process takes time.

"It feels safe here," says Asim. "No one goes into too much detail about their past. I've known I was gay since I was 14. I came out to friends but hid it from my family. We have a traditional Middle Eastern background, so I always knew they wouldn't react well. Then, a couple of months ago, my brother went through my phone and saw some evidence of me being gay. He told them and they sat me down and confronted me.

"They said that, basically, I had two options: I could stay with them and get 'cured' or I could leave. They wanted me to stay but I knew I couldn't live like that. My family aren't bad, they just don't know any better," he says.

As one young person leaves, you can be sure another will quickly replace them. There is no shortage of LGBT young people being shunted out of their family home.

"Since I started here 18 months ago, we are seeing a quarter-on-quarter increase in the need for services," Bowyer says.

In 2010-11, AKT supported 149 young people in London and 85 in Manchester with one of its services. In 2012-13, the number had doubled in London and risen to 152 in Manchester. The amount of mentoring time given in London tripled to more than 1,500 hours over the same period. Overnight shelter provided across both cities rose from 2,434 nights in 2011-12 to just over 3,000 last year.

Bowyer is amazed at the mental strength of AKT's young people. "I've not seen this kind of resilience before, and I've worked in various parts of the homeless sector," she says. "Sometimes everything has been so stressful for them that the reality of their situation doesn't hit them until their head has hit the pillow. Some of them have had really awful lives, but there's something in them that just keeps them going. They are a pleasure to help. We want to give them a better chance in the world."