Jimmy Carter was cheered when he visited Newcastle with Jim Callaghan. Bill Clinton was lauded in Northern Ireland. But it is more usual, at least with more consequential holders of the office, for American presidents to be told by European demonstrators to go home.

The postwar history of our continent would be different and less benign if the United States had heeded that message. His office, and the system of collective security from which we benefit, would be justification enough to welcome President Bush's visit to London this week. But there is an additional reason peculiar to the Bush presidency. For all Bush's verbal infelicity, diplomatic brusqueness, negligence in planning for post-Saddam Iraq, and insouciance regarding standards of due process when prosecuting the war on terror, the world is a safer place for the influence he has exercised.

When Bush ran for president in 2000 he was an isolationist advocate of scaling back America's overseas commitments. But after 9/11, he was right in not interpreting the attack as confirmation that America was stirring up trouble for itself. The theocratic barbarism responsible for the attack on the Twin Towers was driven not by what America and its allies had done, but by what we represented. In the words of Osama bin Laden, illegitimately appropriating for himself the mantel of Islam, "every Muslim, the minute he can start differentiating, carries hate toward Americans, Jew, and Christians".

The most fundamental decision in western security policy in the past seven years has not been the overthrow of Saddam Hussein. It has been the recognition that the most voluble adversaries of western society are not merely a criminal subculture, and still less an incipient liberation movement. Rather, they are a reactionary, millenarian and atavistic force with whom accommodation is impossible as well as intensely undesirable.

The grand strategy pursued by the US under Bush has overestimated the plasticity of the international order, but it has got one big thing right. There is an integral connection between the terrorism that targets western societies and the autocratic states in which Islamist fanaticism is incubated. Bush is culpable for much that went wrong after the overthrow of Saddam, but the outlook for Iraq has changed fundamentally owing to his decision to appoint General David Petraeus and pursue a confrontational strategy with al-Qaida in Iraq.

Bush was wrong, in his 2002 state of the union speech, to speak of an "axis of evil" connecting Saddam, Iran and North Korea – not because he overstated these actors' malevolence but because they were not a homogeneous threat. Two of them remain potent and unresolved problems. But little can be accomplished in restraining North Korea's bellicosity without the active support of China, and at least the Iranian regime has faced a united international front in constraining its nuclear ambitions. Whoever succeeds Bush as president will benefit from some decisions well conceived if often badly executed. So will America's allies.