The legend that Shannon Lee grew up with goes like this. In 1971, when her father Bruce Lee was at the height of his Hollywood kung fu fame, he tried to develop a TV show called The Warrior. It followed a Chinese immigrant with martial arts skills who gets caught up in the Tong wars of the late 1800s, the violent clashes between rival factions in Chinatowns across America. Our hero finds himself journeying through the wild west.

Lee envisaged himself in the leading role, and pitched The Warrior to a US studio, which told him that America wasn’t ready for an Asian leading man. A year later, Warner Bros released Kung Fu, which followed a Shaolin monk with martial arts skills journeying through the wild west. The monk was played by a white actor: David Carradine.

So Lee, frustrated with Hollywood, returned to Hong Kong, where he made Fist of Fury, Way of the Dragon and Enter the Dragon, the three films that secured his legend and saw his image – washboard stomach, raised fists, ready for action – plastered across posters in student flats the world over. That legend only grew after the star died suddenly in 1973 at the age of 32.

Facebook Twitter Pinterest Bruce Lee in Enter the Dragon. Photograph: Moviestore/Rex/Shutterstock

“We grew up with this story,” says Shannon. “That my father created this show and was told he couldn’t star in it because American audiences wouldn’t accept a Chinese leading man. If you were to speak to Warner Bros, I’m sure they’d say they had the idea [for Kung Fu] prior to my father’s pitch. However, they have a lot of similarities, so you know…”

While Shannon, now 50, never doubted her dad’s story, it was only in her mid-30s that she began to uncover just how detailed his pitch for the show was. “In late 2000,” she says, “I agreed to take over looking after my father’s legacy from my mother and she sent his things to me in LA. There were boxes and boxes of writing. While I was going through them, I came across the treatment for The Warrior. It was this revelatory moment, ‘Wow, it really does exist!’”

At that time, Shannon “didn’t feel ready” to tackle a TV show, so she put the notes back in the box, where they stayed – until she received an out-of-the-blue call from Fast and Furious director Justin Lin. “He said, ‘I just wanted to ask you a question – I’ve always heard this story that your dad had an idea for a TV show. Do you have any idea what happened to it?’ I laughed and said, ‘Yeah, I have it right here.’”

Facebook Twitter Pinterest Warrior, the new Cinemax series starring Andrew Koji, right. Photograph: HBO

Lin brought on board Jonathan Tropper, the co-creator and writer of superviolent smalltown crime saga Banshee. He expanded Lee’s original idea into a 10-part series. The result, Warrior, is a ferociously paced mix of period drama and martial arts moves – like the gangster hit Peaky Blinders transported to 1870s San Francisco and given an extra jolt of adrenaline. It’s the sort of show where people are only ever one fantastically choreographed move away from a knockdown fight, where sharp one-liners are delivered with a knowingly raised eyebrow, and the plot moves with a pulpy momentum.

While there are lovely turns from Jason Tobin as sly gangland heir Young Jun and Olivia Cheng as a fictionalised version of the notorious Ah Toy, reportedly the first Chinese madame in San Francisco, the whole thing is held together by an ice-cool turn from British actor Andrew Koji as Ah Sahm, the wandering warrior who arrives in San Francisco intent on tracking down his missing sister, only to find himself caught up in a burgeoning gang war. Koji told the US website Collider that he was influenced by Lee’s own philosophy: “If I just wanted to look good because I want to be flashy and fancy, it would be what Bruce Lee would’ve called an empty, hollow mess.”

For Shannon, the show’s executive producer, it’s the perfect realisation of her father’s dream. “Not only have they righted the original wrong and cast an actual Asian in the lead role, they’ve created this whole complex world that’s dynamic and funny and interesting and sexy. My father’s original treatment was very modern and this really reflects that. It feels true to his spirit.”

Facebook Twitter Pinterest Shannon Lee and her father, who died when she was four.

Shannon was four when her father died and admits she has struggled with separating the screen star from the reality. “I used to think I was a little crazy because I’d feel like I knew my father so intimately. And I’d think, ‘But how can I because I was so young when he died.’ Then I realised my strongest memory of him was the feeling of him. When I think of him, it’s his energy and his love, the security I felt around him, his playfulness. In a strange sense, even though I lost him so young, I feel as though I know him in a very pure way, because my memory is uncluttered by resentment over the time he punished me by sending me to my room or told me not to date that boy.”

It wasn’t always that easy. As children, she and her older brother Brandon struggled to cope with their father’s memory – in particular with kung fu. “There was lots of grief and pressure around it, so my brother and I didn’t study it at all,” says Shannon. “We came round in our teens because martial arts was so important to my father and we felt we’d have a closer connection if we took it up. I don’t really practise it so much now.” She studied jeet kune do, the method made famous by Lee, before she took up taekwondo, wushu and kickboxing to prepare her for acting roles in such martial arts films as Enter the Eagles and High Voltage.

These days, she happily tends her father’s flame, overseeing The Bruce Lee Foundation, which was founded in 2002 to spread his blend of jeet kune do and motivational philosophy. “My father was a self-educated man with this inner energy and I think that’s what speaks to people.” Lee does seem to have been ahead of his time: his vegetable-heavy, dairy-free diet and motivational sayings would be all over Instagram today. Such sayings included “Showing off is the fool’s idea of glory” and “If you love life, don’t waste time, for time is what life is made up of”.

Facebook Twitter Pinterest ‘They righted the original wrong and cast an actual Asian in the lead role’ … Andrew Koji as Ah Sahm in Warrior. Photograph: HBO

Does it feel hard knowing that so many people feel they have a claim on her father? “For sure. It’s always funny when I meet people and they say, ‘Oh, did you know that Bruce would do this or that?’ And they’re fans, they didn’t know him even if they think they do.”

When we speak, she has yet to see Quentin Tarantino’s Once Upon a Time in Hollywood, which depicts her father as little more than comic relief – a dime-store philosopher whose hands “are registered as lethal weapons” as he solemnly informs Brad Pitt’s stunt man. “In a perfect world, I would have loved them to consult with me, but they didn’t,” she says. “I do know Mike Moh [the actor who plays Lee]. He’s a lovely guy and I have nothing against him. But what I’ve seen suggests they’re poking fun at my father. I understand there’s the chance to be irreverent and that’s fine, but I hope it’s not negative.”

Shannon later sees the film and hits out at it, telling the Wrap her father “comes across as an arrogant asshole who was full of hot air, and not someone who had to fight triple as hard as any of those people did to accomplish what was naturally given to so many others”. She added: “It was really uncomfortable to sit in the theatre and listen to people laugh at my father.”

Facebook Twitter Pinterest Shannon Lee in Enter the Eagles Photograph: Everett Collection Inc/Alamy

Perhaps she should have seen it coming. After all, during her father’s lifetime, his appeal was continually misinterpreted by Hollywood. “They thought of him as a one-trick pony. They never understood the combination of skill, thoughtfulness and talent behind the image.” It took Hong Kong and the freedom it provided to reveal the real Lee, the one the world celebrates today. “Hollywood gave him one direction all the time whereas in Hong Kong, with Way of the Dragon, he wrote that film, he directed it, he starred in it, he choreographed it. He had full creative control. He was always writing – he had so many ideas.”

The success of Warrior, which has already been picked up for a second season, has a bittersweet resonance for Shannon, whose brother Brandon was on the verge of stardom when he died, aged 28. It happened during the filming of cult movie The Crow, when a prop gun malfunctioned on set. “Brandon was such an artist,” she says. “He had the ability to become a major star and if that had happened he would have been one of the first actors of Asian descent to cross over.” She sighs and there is a long pause before she adds quietly: “I think about my father and my brother all the time. I miss them the whole time.”

Another pause and then a laugh. “But in my own way, I also communicate with them – often quite energetically. And I feel protected and guided and overseen by them. On Warrior, people would come up to me the whole time and say, ‘This feels great. I feel like your father is here with us.’ I would say, ‘Believe it’ – because I really feel that he was.”

• Warrior is on Sky One on Tuesdays at 10pm. The whole series can be viewed at Sky On Demand and NowTV.

• This article was amended on 6 August 2019. In an earlier version, a picture caption incorrectly said Warrior was an HBO show.