Canaletto’s ‘The Bacino di San Marco on Ascension Day’, c.1733, alongside Turner’s ‘Bridge of Sighs, Ducal Palace and Custom House, Venice: Canaletto Painting’, 1833

In 1799, 24-year-old JMW Turner visited the exquisite picture collection belonging to John Julius Angerstein that would later form the nucleus of London’s National Gallery. Halfway through the tour, the elderly merchant found the young artist standing awkward, agitated and in tears before Claude’s “Seaport with the Embarkation of the Queen of Sheba”. Asked why, Turner burst out: “Because I shall never be able to paint anything like that picture!”

For the next 50 years, Turner’s debate with Claude determined the development of the most revolutionary art ever produced in Britain. An arch-romantic as well as an abstract painter avant la lettre, conveying emotion through pure colour and light, Turner was a radical who engaged in life-long dialogue with both Mediterranean classicism and Dutch naturalism. He took apart the order and geometry of the former, and heightened the realistic effects of the latter, to assimilate them into his own dramatic art of 19th-century conflict and pessimism.

This story, though familiar on the page, has never been unravelled in an exhibition. Now Tate Britain’s Turner and the Masters does for Turner what last autumn’s Paris show Picasso et les Maîtres did for the pioneer of modernism: it illustrates in the liveliest terms, canvas by canvas, how a groundbreaking artist proceeds through a deconstruction of the past that is part homage, part rebellion, leading often to triumphs but sometimes to failures that are as revealing.

Claude’s ‘Seaport with the Embarkation of the Queen of Sheba’, 1648, and Turner’s ‘Dido building Carthage’, 1815

The core of the show is a series of magnificent juxtapositions. Some were dictated by Turner – “Dido Building Carthage”, the sun-drenched landscape fraught with omens of doom, hangs alongside the serene Claude “Seaport”, which it eventually answered, as stipulated in Turner’s will. But most unite his works with their models as Turner could never have dreamed. “Crossing the Brook”, a Devonshire landscape in Italianate classical style, based on a print of Claude’s “Landscape with Moses Saved from the Water”, joins Claude’s original, visiting from Madrid’s Prado. “Port Ruysdael”, the stormy/delicate seascape described by Ruskin as “absolute perfection – the best grey sea piece ever painted by man”, comes from Yale to outdo the churning effects of its inspiration, Ruisdael’s “Rough Sea at a Jetty”. Canaletto’s crisp “The Bacino di San Marco on Ascension Day” hangs alongside Turner’s fantastical, misty depiction of the Venetian artist at work, “Bridge of Sighs, Ducal Palace and Custom House, Venice: Canaletto Painting”, and “Depositing of John Bellini’s Three Pictures in La Chiesa Redentore”, a late miracle of dissolving light on broken surfaces, privately owned and unseen in Britain since the 1960s.

This is a crowd-pleasing show that also has a hard-hitting contemporary agenda. Throughout, a visually splendid array of works by Titian, Rubens, Rembrandt and Watteau balances attempts to reposition Turner as one of us – a self-conscious, competitive artist, haunted by a sense of belatedness, who made pictures about pictures: a postmodern player.

The argument begins compellingly. In 1801, the Duke of Bridgewater commissioned from Turner a companion to Willem van de Velde’s “A Rising Gale”. That painting and Turner’s “Dutch Boats in a Gale” hang together here for the first time in more than a century, and show that from the start Turner set out to rival the past. His pendant, considerably larger, reverses van de Velde’s composition, enhances contrasts and injects dynamic movement, as if his ships are on a collision course. Its painterly bravura made the work the star of the Royal Academy exhibition and the first of 10 grand sea-pieces, increasingly violent, that saw Turner acclaimed as “best of the Dutch marine painters”.

Turner next took on Poussin. “Chateaux de St Michael, Bonneville, Savoy” reworks Poussin’s coolly formal “Landscape with a Roman Road”, softening its austere straight avenue with shadows, permeating the scene with a golden light under which Turner’s track disappears into the mountains: Poussin revisited through nature.

Three years later, “The Deluge” was the head-on challenge. Mesmerised at the Louvre by Poussin’s celebrated rendering of the biblical story, Turner produced a version that whips up and re-orchestrates Poussin’s design in a louder key. Sweeping lines criss-cross the canvas in crashing waves; water, matter, bodies, sky, illuminated by a sinister red sun, build into a tumultuous continuum.

That vortex-like composition was repeated, deepened, elaborated, simplified throughout Turner’s career: in the annihilating blaze sucking up waves, boats, and men, as the port itself seems to crumble, in “Regulus”, smeared with thick, glaring white to evoke the fate of the Roman general whose eyelids were cut off by the Carthaginians; in the overlapping loops and conflicting diagonals of “Snow Storm” – depicting a tempest that Turner claimed to have endured, lashed to the mast of the steamboat “Ariel” off Harwich; in the dissolution of buildings, water, people in iridescent swirls of light in “Mercury sent to admonish Aeneas”.

The writer William Hazlitt called such works “pictures of nothing, and very like”. Modern as Claude or Poussin can never be, both as near-abstract paintings of pure sensation, and for their pessimistic vision of man sucked to his destiny in a meaningless whirlpool of history, they are nevertheless rooted in classical structure and in a feeling of lived experience. “Every look at nature is a refinement upon art,” Turner wrote. Significantly, along with “Dido Building Carthage”, he chose to will to the National Gallery “Sun Rising Through Vapour: Fishermen Cleaning and Selling Fish”, referencing at once the sublime sun and the earthy banality of the fishermen, the world beautiful yet mundane, in a painting bearing witness to his mastery of both Dutch realism and Italian idealism.

Tate tells this story commandingly. I am less persuaded by its diversions: Turner’s narrative paintings are clumsy embarrassments at best, never mind when set against Veronese or Titian. In “Venus and Adonis”, Turner’s goddess of love, partly obscured by her swain, is a headless chunk of flesh. “Jessica” resembles, as the Morning Chronicle complained, “a lady getting out of a large mustard pot”. Watteau makes everyone else look over-muscular; Turner’s fêtes are heavy-handed, and his most Watteauesque, “England: Richmond Hill”, is inexplicably excluded. “Pilate Washing His Hands”, unwisely shown alongside Rembrandt, is an over-bright, unreadable mess of lumpen figures. Ruskin attributed these to Turner’s childhood as a Covent Garden barber’s son, growing up with cabbages and tomatoes, buckets and barrows. His escape was running down to the Thames, fixing forever the play of light on water, the drama of riggings and masts, as his visual imprint.

Here he was unbeatable. A Tate coup reunites the marine painting “Helvoetsluys”, from Tokyo, with Constable’s “The Opening of Waterloo Bridge” for the first time since they appeared together at the Royal Academy in 1832. In response to Constable’s bigger, more luxuriant work, Turner on varnishing day added a single daub of red lead, fashioned into a buoy, on his grey sea. Its intensity, vivid against the coolness of the rest of the picture, made “Waterloo Bridge” look weak and fussy. “He has been here,” said Constable, “and fired a gun.” In this wonderful show, one hears the firing of many such guns.

‘Turner and the Masters’, Tate Britain, London SW1, to January 31, tel +44 (0)20-7887 8888; Grand Palais, Paris, February 22-May 24; Prado, Madrid, June 22-September 19