Turnbull hosted BBC Breakfast for 15 years, in both London and later Salford, and also fronted Songs of Praise and game show Think Tank.
His family said: "Bill will be remembered by many as a remarkable broadcaster who brought warmth and humour into people's homes".
In 2018, he announced he had been diagnosed with prostate cancer.
After leaving the BBC in 2016, he joined Classic FM, and continued presenting shows on the network until recently.
His family added: "Following a challenging and committed fight against prostate cancer, Bill passed away peacefully at his home in Suffolk surrounded by his family on Wednesday, 31 August."
They said he received his diagnosis in 2017, and praised his "outstanding medical care" from the Royal Marsden and Ipswich Hospitals, St Elizabeth Hospice and his GP.
"He was resolutely positive and was hugely buoyed by the support he received from friends, colleagues and messages from people wishing him luck.
"It was a great comfort to Bill that so many more men are now testing earlier for this disease," they said, calling him a "wonderful husband and father to his three children" and a devoted Wycombe Wanderers fan and "an ever-aspiring beekeeper".
Turnbull was a favourite with viewers on BBC Breakfast thanks to his gentle presenting style, which was well suited to early morning broadcasting.
During his time on the show, he worked alongside Sian Williams, Susanna Reid, Kate Silverton, Natasha Kaplinsky and Louise Minchin.
His former colleagues Naga Munchetty and Charlie Stayt, who were presenting Thursday's programme, paid an emotional tribute just after the news was announced.
Munchetty said: "As a team we are in shock this morning. He cared so much about this programme and the audience. I did my first ever shift with him on Breakfast."
"His energy was amazing. He came into this programme and threw everything at it. He was funny; he was a brilliant journalist. He loved this programme and he loved serving you, the audience," she said.
Stayt added that Turnbull "didn't take himself too seriously".
The programme's editor Richard Frediani confirmed on Thursday that Friday's Breakfast would be "a celebration of Bill's life on and off-screen". He noted: "Bill Turnbull loved our viewers and our viewers loved him."
On his last BBC Breakfast show, Turnbull himself talked about some of the moments he remembered most.
"I remember nearly getting into a fight with a ventriloquist's dummy named Bob," he said.
"Seriously it nearly came to blows, although that was more me than him."
He added: "And there was a moment where I actually wore a sweater made from dog hair with a bow on the front. It was alright but it was very, very warm and I just couldn't get the stuff off me for weeks afterwards."
Williams told the BBC he "mentored a lot of young journalists and a lot of people in the industry... and that was very important to him", adding he had "very high standards".
She spoke of their relationship, saying: "We never bickered... in 30 years of friendship", and said she would remember them "laughing so much we couldn't really keep control of our faces... and then we realised we were on television - we laughed a lot".
Minchin described him as "a brilliant journalist", adding "most of all he was great fun".
BBC Breakfast's sport presenter Mike Bushell also paid a glowing tribute to his "dear friend" and "mentor" with the "incredible talent".
Reid, now on ITV's Good Morning Britain, added: "I feel lucky to have worked with him, and he taught me everything. But above all, he was devoted to his family and I am heartbroken for them."
Another former Breakfast colleague, Channel 4 presenter Steph McGovern, posted she was "absolutely gutted", and said: "I learnt so much from him. And we had some cracking arguments about how you should pronounce words like 'poor'. We all loved him."
Channel 5 news presenter Dan Walker said it had been "an honour" to replace Turnbull on the Breakfast sofa, saying: "He was full of brilliant advice and it was clear just how loved he was by his colleagues and the audience," while Breakfast presenter Jon Kay called Turnbull "an absolute gent".
Turnbull appeared as a contestant on an early series of Strictly Come Dancing, finishing in sixth place with his partner Karen Hardy.
Hardy, who was in New Zealand, cried as she paid tribute to her friend, calling Turnbull "such a gentleman, such a mentor".
She said meeting him was "life-changing" - he was her first partner on Strictly - and "he made TV magical, he told me what it was about", she said, adding he would help critique her appearances on Breakfast.
"If he wasn't on it, I knew he'd be watching..." she said, adding he would text her with tips including how to sit and speak clearly for the cameras.
BBC Director-General Tim Davie called Turnbull "a much-loved and respected broadcaster and journalist - not just by viewers but by all those lucky enough to have worked with him".
"He always struck the right tone, no matter what the story. Warm, wise, professional and caring, he will be much missed by us all," he said.
Culture Secretary Nadine Dorries described "an incredibly sad loss of a broadcasting legend", praising his "special charm and wit" and his tireless work to raise awareness about prostate cancer.
BBC Radio 4's Today programme presenter Nick Robinson described Turnbull as "a very, very dear friend and an extraordinary broadcaster."
"There was a warmth to his broadcasting which people who watch Breakfast television every day knew," he said. "But perhaps what they forgot was what a bloody good journalist he was. This was a man who'd been a correspondent in Washington, who travelled 30 countries. He'd been in Moscow, he'd covered wars, he'd reported on the troubles in Northern Ireland."
He added: "And that combination of a razor-sharp intellect, wit, humour and humanity, came out every day when he was on Breakfast. It came out every day when he was a reporter on this programme. And listeners to Classic FM will have heard him present beautifully as well his love of music."
Robinson also recalled how Turnbull and his wife [Sarah] had met on the Today programme, where they had worked.
Broadcaster and writer Monty Don called Turnbull "a thoroughly decent, good man", adding: "We shared an enthusiasm for bees and were both patrons of Bees for Development charity."
Prior to working on Breakfast, Turnbull also worked as a reporter on BBC Radio 5 Live and BBC News 24, as the news channel was then known.
In recent years, Turnbull guest presented episodes of The One Show and reunited with Reid for several editions of ITV's Good Morning Britain.
After his cancer diagnosis was made public, he fronted a documentary for Channel 4 called Bill Turnbull: Staying Alive, in which he explored the use of cannabis oil for medicinal purposes.
Prostate Cancer UK said that after Turnbull announced he had the illness in March 2018, their specialist nurses experienced their "busiest day on record" and they saw a huge rise in web traffic of men looking for information and support.
In November 2017, Turnbull took part in a celebrity edition of The Great British Bake Off - which was in aid of a charity, Stand Up to Cancer. He found out about his prostate cancer during filming.
He said inspiring men to get tested for the illness by publicly revealing his own diagnosis was the "one useful thing" he had done in his life, admitting he was "cross with myself" for the pride he had felt at not visiting a GP in four years.
* It's the most common cancer in men in the UK - an ageing population means more men are developing and dying from the disease
* 40,000 new cases are diagnosed and around 11,000 men die from it each year
* It can develop slowly over years and many men have no symptoms
* Noticeable symptoms include needing to urinate more often and weak flow
* There is no single test for prostate cancer - a blood test, biopsies and physical examinations are all used
We explain what warning signs to look out for