I would say that the Hostage Crisis in Iran was a turning point in American politics for a couple of reasons. The first was that it illuminated President Carter's weakness in the eyes of many American voters. When Carter delivers a speech about the "crisis in the American spirit," it seemed that he inadvertently described his own political narrative. The Hostage Crisis in Iran revealed how much of a crisis America faced around the world. The sight of Americans helplessly paraded around in blindfolds helped to confirm to Americans that President Carter was not able to direct American interests around the world with strength and decisiveness. Carter's muddled support of the Shah at a critical time in Iranian politics further cemented this legacy. His failure to fully understand the Iranian Revolution as well as the political climate in Iran did much to undermine his stature in the eyes of the American people. At the same time, his Republican opponent, Ronald Reagan, was able to use the perceived weakness out of the Iran Hostage Crisis and parlay it into a larger narrative. When Reagan stares into the camera and asks Americans if they are better off four years ago than they are now, the vision of Americans held hostage in Iran does much to guarantee their answer and devalue President Carter.
I cannot see how the Hostage Crisis in Iran is only a (small) thorn in Carter's reelection campaign. The lack of intelligence on the ground in Iran and the lack of stature in the region helped to enhance the American public's idea that the Carter Administration was in over their heads and could not guide the country. The fact that President Carter could not turn to any ally in the region to help and that his own credibility with both the Iranians and the Americans was in the same political ballpark helped to make the Hostage Crisis a political liability. The inability to change the narrative and change the political dialogue that arises from the Hostage Crisis in Iran is what makes President Carter an incumbent President fighting for credibility with the body politic. In such a condition, the crisis was not a thorn. It was a significant turning point in American politics because it helped to usher in the Reagan Revolution, which was perceived to be the exact opposite of what Carter was offering.