The upstairs rooms at the Royal Academy are crisp and boxy – like a tidy argument. The curators of this show try to do something similarly neat and compacted with painting in America during the 1930s, tidying it away into themes and moments. It doesn’t work. Does that matter, though? No.

This is not a show of great paintings by a small number of artists. It is a show of highly significant paintings by the many who were struggling to define the nature of an America that was coming into being during a momentous decade. It's an unruly show from first to last, wildly stylistically inconsistent – in fact, it tries to make its own rules as it goes along, looking wistfully over its shoulder at Europe one moment, wrestling with its own nostalgia-soaked version of American pastoral another.

Edward Hopper, 'Gas', 1940 (The Museum of Modern Art, New York/Scala, Florence) (Photo © 2016. Digital image, The Museum of Modern Art, New York/Scala, Florence)

Even when it gives a cold, steady, fact-based stare at industrialised America in the paintings of Charles Sheeler, it doesn't quite know how to cope with it, how to make sense of it, how to turn it into a current reality. The reality – whatever strange something it is that is underpinning this vision – seems to spin away, to blur off into a strangely frozen unreality, like a turbine out of control. Look at how Philip Guston captures the idea of the aerial bombardment of Guernica, representing it in a swirling tondo of stricken figures, weirdly extended physically, wrenched awry, limbs straight out of Parmigianino.

In this respect, the show feels like journalism at its best, forever trying to make sense of the sheer, no holds barred cacophony of a ceaselessly unpeeling story. Styles are tried on and shucked off like unwanted clothes. There's an anxiety about using paint at all. See how the marvellous, near airborne structure of the George Washington Bridge seems to melt away like over-long masticated gum in a painting by O Louis Guglielmi. There is an anxiety here, everywhere, a blundering through, a reaching out, often tentative. Is the slow deliberation of paint quite equal to the occasion?

Jackson Pollock, 'Untitled', c. 1938-41 (© The Pollock-Krasner Foundation ARS, NY and DACS, London 2016)