"Freedom of the press in Britain means freedom to print such of the proprietor's prejudices that the advertisers don't object to"
Americans like to believe their press is neutral. In the UK, hardly anyone bothers to maintain this pretence, and the press are openly engaged as political factions. The press circulation is much larger than the membership of the political parties, and has an impact beyond its actual numbers in terms of influencing opinion.
The three largest papers are the Sun (owned by Murdoch's News Corp), the Daily Mail (owned by Viscount Rothermere), and the Times (News Corp again). All have a right-wing anti-Europe stance, and will happily print inaccurate articles about Europe. Boris Johnson got a particularly bad reputation for this while working as a journalist before he was a minister. He is still employed as a journalist, and is in fact paid more for that than his role as a Minister of the Crown.
The press benefit from controversy: it sells papers. Readers enjoy outrage bait that panders to their prejudices; a pre-internet version of the "fake news" problem.
Sidebar: the Spiked/LM axis
There is a small group of people who used to write for Living Marxism until it was sued out of existence for denying the Bosnian genocide, then formed a successor Spiked!. They seem to be surprisingly influential in the commentariat despite their small size, and despite the intellectual incoherency of having gone straight from revolutionary communism to libertarianism without passing through common sense on the way.
One of these people is Brendan O'Neill. Another is Claire Fox, now Baroness Fox of Buckley.
BBC Question Time
A similar story prevails here. What ought to be a discussion show has succumbed to the temptation of getting on the most extreme guests possible, provoking highly strung arguments, and letting planted audience members inject outrageous questions.
Edit: another example of the revolving door: the man who ran the BBC's political output during the Brexit campaign was later appointed as Theresa May's director of communications and is a "hard Brexiteer".
Culture: The Establishment
The leaders of the country overwhelmingly come from a very narrow educational background: private school followed by Oxford PPE. This produces people with a bulletproof opinion of their own correctness and practice in intellectual bullying.
Culture: Ruins of the Empire
This mostly manifests as a belief that Britain can "punch above its weight", and therefore does not need to engage in international compromise. While the influence of this in negotiations with Europe is obvious, I would argue that it also prevails internally. (This is easily a graduate-thesis size investigation!). It seems to me that the country is run centrally as a tiny empire. Councils have little power and little funding autonomy; council tax rates can be capped centrally, and they are dependent on "block grant". The relatively recent devolved assemblies also get little respect from Westminster; the Scotland Office and Wales Office still exist despite seeming redundancy. The Scotland Office has a substantial budget for anti-independence campaigning.
The situation is even worse in Northern Ireland, land of No Surrender, where there has been no government for about two years following its collapse over a fraud scandal. Compromise is even more foreign there. People tend to forget that the UK had a live-fire civil war in living memory, but I think that matters to the uncompromisingness.
Systems: First Past the Post
The voting system at Westminster discourages coalition or minority governments, so there is no tradition of sound coalition-forming by seeking consensus.
Edit: as a piece of metacommentary, this is the first time I've noticed an answer getting significant numbers of upvotes and downvotes. I am aware that it can be considered opinionated, but I think that's a symptom of the polarisation that leads to no consensus. Politically engaged people disagree on increasingly basic things.