In an interview with the Standard, Mr Turner said he would “demand” a seat at the table of the Labour Party and called on leader Sir Keir Starmer to listen to “ordinary working people” to rebuild trust.
“I want Keir and the rest of the party to come out with me on my tour,” he said while on a visit to Dagenham.
“I want them to listen to our members, I want them to walk in our communities, I want them to come and meet my mates down the pub.
“I want them to rebuild trust and have an ambitious programme that people can buy into that motivates and inspires them to come home to vote Labour. That’s the task of Keir.”
He said Scotland was the first so-called “red wall” and added: “If you take people for granted, they’ll walk away from you eventually.”
During the interview, Mr Turner also vowed to be the “greenest” union leader ever and said working class communities must not be left behind in the transition to a green economy.
“We can’t desert communities because it’s no longer fashionable to be doing the work that they’re currently doing now, we have to take people with us on this journey to transition our economy,” he added.
“I look back at the mining communities of the 1980s…and they were just completely deserted. There was no investment in them, there was no transition, certainly not a workers’ transition. And people were just left. We’re not buying that pig in a poke.”
The former London bus conductor also said he would be fighting for a “legacy of furlough” – the unprecedented policy that protected millions of jobs during the pandemic.
“We’re not going to put the genie back in the bottle, we want to have a legacy from furlough – around longer-term protections like they have elsewhere to deal with the peaks and troughs in the economy.”
A “proud Londoner” and Millwall fan, Mr Turner grew up on the now demolished Heygate estate in Elephant and Castle.
He said he had “worked very well” with London Mayor Sadiq Khan during the pandemic to support transport workers but said there was a problem in the capital with regeneration programmes which he claimed too often were an act of “class cleansing”.
“I came out the Heygate, in Elephant and Castle, and that’s completely demolished. That’s a gross act of class cleansing if ever I saw one – people being dispersed across our capital.
“When is anyone ever going to build an average apartment? They all build these luxury apartments, as if nothing else matters and it’s probably, what, £600,000 pounds? That used to be the home of my community.”
Mr Turner first joined a union as a teenager before being elected as a shop steward for London buses and joked it was because he was the “gobby one”.
He added: “I joined our union as a 19-year-old at Walworth bus garage. The day I joined, the shop steward put their arm around me and they gave me an opportunity to become a different person. I was in a difficult place as a young kid in London.
“I could have gone in different directions. The union gave me an opportunity to become the person that I am now and I’m forever grateful for that.
“I’ve spent 39 years now putting my arm around people, either individually or collectively.”
Against Mr Turner is Gerard Coyne, seen as the moderate candidate, and Sharon Graham, who has the support of the Socialist Workers Party. Unite’s current general secretary Mr McCluskey is backing Mr Turner in the race.
At stake are millions of pounds of grants and donations funnelled from Unite’s accounts into political campaigning, as well as the union’s influence over the selection of future MPs and in policy votes.