Learn English – Are the origins of the idioms “on the fly” and “just wing it” related

This site says that it comes from a 1728 play. To the extent that it's known nowadays, though (it's not really prevalent anywhere, as far as I know), it's due to its use in Lewis Carroll's Alice in Wonderland.

As others have said, it's a general term conveying the idea of one by one.

I'd like to add, though, that at turns may have been used because there are two things that could be said to have come one after another: the candidates, and their gaffes. So, in no particular order: Bachman was perceived as ridiculous, Cain was perceived as self-destructive, Huntsman was perceived as ineffectual, etc.

The author probably didn't want to be so specific, partly because the correlation between candidates and gaffes isn't as clear-cut as I've indicated here, and partly because a single candidate may have compounded the said gaffes atop each other in a series of political missteps. So, an efficient way to convey this sentiment is to state the generality:

His declared rivals were at turns ineffectual, ridiculous, or self-destructive...

leaving the reader to work out the details of who stumbled when, and how each one stumbled. Meanwhile, the author can move straightaway to the main point of the sentence, which is that Romney had "a fairly easy, if occasionally fraught, path to the nomination."

I'd conclude that at turns is not idiomatic, but instead an efficient way to describe a sequence of people independently making a series of mistakes.