DES MOINES, Iowa—Attacks on Jeb Bush are ramping up even before he enters the 2016 presidential race, with fellow Republicans questioning everything from his conservative bona fides to whether he is prepared for a long campaign to become the party’s nominee.

Mr. Bush’s rivals, 10 of whom joined him for a Saturday night fundraising dinner for the Iowa Republican Party, have latched onto the former Florida governor’s shaky responses last week about the wisdom of his brother’s entry into the Iraq war, ensuring the issue will dog him for the foreseeable future.

It highlights the core challenge of Mr. Bush’s expected formal entry into the campaign: a family name that has given him outsize attention and access to an unparalleled fundraising network that at the same time could make it difficult for him to forge a political independence.

It also shows that even though Mr. Bush’s political standing is weaker in Iowa than elsewhere, he remains the focus of the race, with the other presidential hopefuls implicitly defining themselves against him.

Kentucky Sen. Rand Paul, who has positioned himself as the only likely GOP candidate not presenting an aggressive overseas military posture, suggested to the audience of 1,300 Republicans at the state party’s Lincoln Day dinner here that Mr. Bush would mount another unpopular war.