· Poll shows rising anxiety in both US and Europe · Iran now seen as country posing the gravest threat

The Bush administration's claim that Americans are now safer from terrorism has been undermined by a poll yesterday showing a significant jump in the number of Americans and Europeans concerned about Islamist extremism and other global threats.

The survey, published days before the fifth anniversary of 9/11, is at odds with a Bush administration document on terrorism on Tuesday that insisted: "America is safer, but we are not yet safe."

The annual Transatlantic Trends survey, conducted in the US and 12 European countries, records that concern among Americans has risen from 72% last year to 79% this year, and among Europeans from 58% to 66%. The biggest jump in concern about Islamist fundamentalism is in the UK, up 22 points.

Concern is expressed in the poll on both sides of the Atlantic, not only about global terrorism and Islamist fundamentalism but the Iraq war, immigration, global warming, the Iraq conflict and confrontation with Iran. The poll finds a high level of support both in the US and Europe for military intervention to stop Iran securing a nuclear weapon capability should diplomacy fail.

Transatlantic Trends, funded by bodies including the German Marshall Fund of the US, conducted phone interviews with 1,000 people in June in 13 countries: the US, the UK, Germany, France, Italy, Netherlands, Poland, Portugal, Slovakia, Spain, Turkey, Romania and Bulgaria.

The survey records a steep decline in European support over the last five years for US leadership in world affairs. The more inclusive approach towards Europe taken since last year by Condoleezza Rice, the US secretary of state, has failed to reverse this decline.

In 2002, 64% of Europeans took a positive view of the US leadership in world affairs but this has dropped to 37%. Only in three European countries is there a still a majority supporting the US: the Netherlands, Romania and the UK (where opinion is almost evenly divided, 48% positive to 45% negative).

Europeans polled distinguish between the US and Mr Bush. When asked about Mr Bush's leadership in foreign affairs, support drops even further, from 38% who took a positive view in 2002 to only 18% this year.

For the first time since 2002, more Americans (58%) disapprove of Mr Bush's handling of international affairs than approve (40%), reflecting a steady erosion in support for the Iraq war.

Iran has taken over from Iraq as the country viewed as posing the gravest threat. With the confrontation with Iran set to intensify over the next fortnight as the US seeks a UN security council resolution to impose sanctions, an overwhelming majority of Americans and Europeans support a continuation of diplomatic activity to prevent it securing a nuclear weapon capability. Only 15% of Americans and 5% of Europeans regard military action as the best option.

But there is a dramatic rise in support for military action in the event of a complete diplomatic breakdown. In that case, 53% of Americans would support military action and 45% of Europeans. Support for the military action is higher in France, at 54%, than in the UK, at 46%.