Sarah Palin's hopes of being the Republican nominee to take on Barack Obama next year are suffering from the fallout from the Arizona shooting spree, under criticism from both left and right.

Although there is no direct line from Palin to the suspect, the reality is that she is being damaged by the repeated television showings of her having targeted Congresswoman Gabrielle Giffords using a crosshairs image on her website. The footage also shows Giffords warning Palin that her use of such violent imagery would have "consequences".

Palin's chances of announcing a run for the White House dived over the last 48 hours on Intrade, the politically neutral online trade/betting exchange, dropping from around 75% to around 60%.

"The long and the short is she is damaged," Larry Sabato, professor of politics at the University of Virginia, said. "This is obvious even to the base. I do not think other Republican candidates will use it against her. They do not have to because there will be so much commentary. She has identified herself with the most strident elements of the party."

Palin and her small inner circle are reported to be appalled and disgusted that the left has linked her to the shooting. But it is not just the left who will welcome her discomfort. Many senior Republicans do not want to see her run for office.

Sabato said he had spoken to several senior Republicans who believe she is damaged, describing her as one of the additional victims of the tragedy.

"The truth is when you get beyond the party activists, you can't find senior Republicans who want her to get the nomination," Sabato said. "They know she will get beaten by a landslide in the election [against Obama]. They are not stupid."

The left claims the violent rhetoric of Palin and the Tea Party movement has contributed to the present toxic political environment. It cites Palin's use of language such as her call to supporters to "reload" after losing the health care debate.

The Tea Party movement responded that it is not clear what the politics of the shooter are, with some claiming he is a liberal "pothead" rather than a rightwing conspiracist. They say the use of warlike rhetoric has long been commonplace in politics, with talk of campaigns and battles.

Mark Meckler, one of the leaders of the Tea Party Patriots, told the Daily Beast website: "To see the left exploit this for political advantage – some people have no conscience. It's genuinely revolting … I think it sinks to the level of evil."

The problem for Palin is that she is caught up in the centre of an ugly debate, not a good place for a would-be presidential candidate who has to win on the centre ground.

David Frum, a former Bush speechwriter, argued that Palin's response – condolences to the victims and a simple denial of any responsibility – was perfectly fair in reply to opponents who suggested the crosshairs contributed to the shooting. But if she wanted to be a leader, she had to go beyond that, he said. He called on her to express real grief and sincere compassion, be visible, engage in the debate, and challenge opponents,

"Palin is splashed by that history [the build-up to the Tucson incident] and the challenge is to find something big and generous to speak of in a larger way. So far, she has opted not to do so," Frum said.

Time is running out for Palin. In theory, she could leave it until towards the end of the year to declare her intention to seek the Republican nomination. In reality, the invitations are already being sent out to participate in debates against other candidates in Iowa and elsewhere round the country.

The date of the first test, the Iowa caucus, is likely to be early next year, but candidates will have had to have visited the state many times before that.

A poll in Iowa as well as a nationwide Gallup one today had Palin trailing, though the figures are largely meaningless this early in the political season other than as a rough guide.

The main significance of the Gallup poll is that it characterises Palin as a polarising figure, with high negative ratings, even among Republicans.

Andrew Sullivan, the conservative blogger, writing in response to Palin's tanking in Intrade, wrote: "There is no way to understand the politics of this without Palin. She has long been the leader of the movement that drapes itself in military garb, that marinates in violent rhetoric, that worships gun culture, that has particular ferocity in the state of Arizona, and that never ever apologises for anything."

He adds: "My hope is that this horrifying momentary conflation of politics, guns and mental illness will lead responsible figures on the right to eschew the path of Palin."

Steve Clemons, a liberal blogger and a director at the Washington-based think-tank New America Foundation, saw Palin as culpable for "the whole brand of lock and load politics, that frontierswoman mystique and a continuation of the Bush Texas-style swagger".

He said a culture had developed in American politics in which people "yell out … and stomp their feet, and act as if brute force is something to be celebrated". He added: "What used to be celebrated is ability to think through politics. What we are celebrating is ignorance. I think Sarah Palin has had an enormous amount to do with this." He predicted Palin and the Tea Party would be forced to the fringes in the near term, but predicted both would prove resilient.

Mark Blyth, a professor of politics at Brown University, was equally hardbitten in his assessment. In a week's time, he said, what people would be talking about would be the jobless market, cutting the deficit and the cost of oil, rather than the shooting.

"Is this going to be the thing that sinks Sarah Palin? Probably not," he said.