The language of privatisation this government parrots so enthusiastically is a language of emancipation. In the "free" market, hidebound public institutions escape the "shackles" of the state. They "tear up" red tape. They "diversify", "innovate" and "liberate" themselves from oppressive controls. "Business is the most powerful force for social progress the world has ever known," said David Cameron recently. "It can help us to smash poverty, raise horizons, drive the innovations, products, services that make our lives better, longer and happier."

He did not understand that the readiest alternative to public money is not the free market. If a public servant needs a fortune fast, he can turn to Russia, China, Saudi Arabia and the other oil-rich dictatorships. Their propaganda budgets can deliver lump sums that spare the harassed bureaucrat from the tiresome need to woo scores or hundreds of private companies and individual donors. All the dictatorships ask in return is that Britain's cultural institutions sell their souls. They are more than happy to haggle.

The British Museum's current Hajj exhibition charts the history of Mecca as a destination for pilgrims with the wariness of a conscript crossing a minefield. The exhibition sticks to the authorised version of "religious scholars". It allows no discussion of the findings of historians of Islam – "true scholars who have read more than one book", as Richard Dawkins puts it – that the traditional account is as much a fairy story as the traditional accounts of Christianity and Judaism. Fear of bombs in the building or of staff receiving the same treatment as Salman Rushdie and Ayaan Hirsi Ali have kept evidence about the Muhammad of history far from public view.

The exhibition goes further than the standard tongue-biting editor or panicked publisher, however. It not only fails to question Islam's foundation myths but augments the myth-making by excluding evidence that might embarrass the Saudi royal family. In a piece for the American arts magazine Guernica, Joy Lo Dico embarrasses other critics by pointing out what was in front of their noses. Saudi Arabia provided exhibits. The Saudi royal family's King Abdulaziz Public Library partnered the museum. HSBC Amanah, a bank that issues sharia-compliant loans, sponsored the show. By negligence or design, nothing in the exhibition offends the Saudi state, which derives legitimacy from its control of sacred sites and income from pilgrims.

You might have thought that of all people the museum's director, Neil MacGregor, would deplore cultural vandalism. The author of A History of the World in 100 Objects would surely deprecate the destruction of buildings of historical significance. He must know that Saudi's monarchical dictatorship has wrecked Mecca with an abandon worthy of the Taliban. It has destroyed the remnants of the 7th-century city, most notably the houses of the prophet, his first wife and Abu Bakr, father of Aisha, one of Muhammad's other wives.

According to the Wahhabi monarchy's puritanical and iconoclastic version of Islam, anything that generates idolatry – images of the prophet, homes associated with him – is dangerous. So medieval Mecca had to go.

The British Museum does not mention the hooliganism. Nor does it mention the occupation by 400 jihadists of the Grand Mosque during the 1979 Hajj and the deaths of hundreds in the gunfights that followed. Saudi Arabia has excised all reference to a battle that prefigured the rise of jihadist terrorism from public discourse and school textbooks. The British Museum can't find space for it, either. It also fails to show pictures of the stampedes, bridge collapses and fires that have claimed the lives of thousands of pilgrims. They might have provided a sombre comment on the Saudi authorities' incompetence.

The museum might say that unless it promised not to offend the monarchy, the Saudis would never have given permission for exhibits to leave the country. But censorship is at its most effective when no one admits it exists. If otherwise admirable institutions were to admit they censored for money or access, the propaganda value of the exercise would vanish. Dictatorships want to scrape an acquaintance and gain virtue by association. Virtue would vanish if everyone knew that the dictatorship had bought silence as part of the package. The museum would lose credibility. The dictatorship would look dictatorial.

Thus, instead of offering a frank conversation, the British Museum waffles to me and other journalists that the destruction of Mecca and the terrorist attacks did not fit into the exhibition's remit. Like an Oxford don, they respond to questions by saying they would love to help but it's not quite their specialism, although how the death of thousands on the Hajj is not a part of the story of the Hajj is beyond me.

Similarly, when the London School of Economics made up its funds by taking money from the Gaddafi crime family. It did not say: "Obviously, after pocketing Libya's cash, we cannot criticise the dictatorship or examine its oppressive methods because the dictatorship would stop paying us if we did." Instead, it pretended that it did not matter that the Libyan dictatorship was a dictatorship and changed the conversation.

Saudi money, even Syrian money, props up the decaying university system. Academics tell journalists that the next great scandal in higher education will be about Chinese influence in the universities, which dwarfs that of the Middle Eastern monarchies and tyrannies. Not that we have a right to criticise vice-chancellors. In Deception, a new account of the attempts by Putin and his pet oligarchs to spy on the west, Edward Lucas explains that the English libel law is not the only weapon they can use to force journalists, thinktanks and analysts to mute their criticisms.

The coerced Russian taxpayer pays for supplements in the Washington Post and Daily Telegraph, filled with puff pieces for the regime. The reader has to stare very hard at the page to find the disclaimer in tiny print that the "news" has not been produced by its journalists. As always, the object of the exercise is to make propaganda sound like truth and special pleading look impartial.

Because the 2008 crash not only shifted power from west to east but from democracies to dictatorships, more cash-starved western institutions will be following the dictators' money. I want the state to fund universities and the arts. As it is not going to, I would prefer the British Museum to charge for entry and universities to increase their tuition fees. It's better than selling themselves in the unfree market.