The Church of England has delivered a withering critique of David Cameron's Middle East policy, describing the government's approach as incoherent, ill-thought-out and determined by "the loudest media voice at any particular time".

The criticisms are made in an extraordinary letter to the prime minister signed by the bishop of Leeds, Nicholas Baines, and written with the support of the archbishop of Canterbury, Justin Welby. Seen by the Observer, it describes the UK's foreign policy as so muddled and reactive that it is "difficult to discern the strategic intentions" of the government's approach to the region.

The letter follows widespread claims that Britain and the west have been slow to respond to unfolding events in Iraq as Islamic State, formerly known as Isis, has imposed its bloody rule across northern Iraq and swaths of Syria.

Cameron is taken to task for failing to develop an effective plan to tackle the spread of violent Islamist extremism from Iraq to Nigeria, where the militant group Boko Haram has terrorised the north of the country. "We do not seem to have a coherent or comprehensive approach to Islamic extremism as it is developing across the globe," the bishop writes.

Cameron is accused of turning his back on the suffering of Christians. The letter asks why the plight of religious minorities in Iraq, such as the Yazidis, seems to have taken precedence. It notes that, though the government responded promptly to reports of at least 30,000 Yazidis trapped on Mount Sinjar, the fate of tens of thousands of Iraqi Christians fleeing jihadists from Mosul, Iraq's second city, and elsewhere appears to have "fallen from consciousness".

Baines asks: "Does your government have a coherent response to the plight of these huge numbers of Christians whose plight appears to be less regarded than that of others? Or are we simply reacting to the loudest media voice at any particular time?" He condemns the failure to offer sanctuary to Iraqi Christians driven from their homes: "The French and German governments have already made provision, but there has so far been only silence from the UK government."

The letter asks why parliamentary questions tabled last month to find out whether the UK intends to offer asylum to Iraqi Christians have still not been answered. Baines says the failure to confront the issue is "something that causes me and colleagues some concern".

Writing in the Sunday Telegraph, Cameron suggests he is inching towards supporting Britain's involvement in a military response to the threat from the Islamic State. He says it is important not to let the controversy and tragedy of the 2003 war in Iraq colour Britain's response to today's crisis.

He writes: "The creation of an extremist caliphate in the heart of Iraq and extending into Syria is not a problem miles away from home. Nor is it a problem that should be defined by a war 10 years ago. It is our concern here and now. Because if we do not act to stem the onslaught of this exceptionally dangerous terrorist movement, it will only grow stronger until it can target us on the streets of Britain.

"We also need a broader political, diplomatic and security response … This threat cannot simply be removed by air strikes alone. We need a tough, intelligent and patient long-term approach that can defeat the terrorist threat at source."

As Britain prepares to give military assistance to the Kurdish peshmerga forces in northern Iraq, a leading security expert has warned that recent military intervention in northern Iraq could play into the hands of Isis. Richard Barrett, former head of counterterrorism at MI6, said air strikes and the decision by Cameron to arm Kurdish forces could escalate the threat against the west, hardening the resolve of jihadists to such an extent that fighters from Isis and al-Qaida might join forces.

Barrett, who spent more than a decade tracking the Taliban for the UN, said the attacks on Isis "feed the narrative that America, the west, is part of the problem". He added: "If the west is part of the problem, then the question is: 'Why don't you attack the west right now?' There's a definite possible downside that action by the US, particularly if it's prolonged, could lead the al-Qaida guys and the Islamic State guys to say: 'OK, let's get back together and do this'."

US intelligence assessments suggest that some jihadists are abandoning al-Qaida affiliates in Yemen and Africa to join Isis. Spy agencies are watching to see if senior al-Qaida leaders are prepared to switch allegiance.

Barrett said the west also needed to confront the dilemma that, even if Isis were defeated on the battlefield, significant problems would lie in store. "They have their own territory to defend now. Even so, they are not going to all die on the battlefield; the many thousands of foreigners ... will go back home if they are defeated with a strong sense of injustice and a strong motivation to carry on the fight. Of course, if they are not defeated, then they will want to spread their rule into their homelands anyway. You sort of lose either way."

The Metropolitan police's senior national coordinator for counter-terrorism, Helen Ball, said the situation in Iraq had evolved to the extent that police were appealing for the girlfriends, wives and mothers of potential jihadists to dissuade the men from travelling abroad.