Walking around this large exhibition - 50 big paintings and a crowded room of drawings, 20 years of work, the artist's first one-man show in a British museum - it is hard to suppress a mounting, suspiciously euphoric feeling that Doig is our modern-day Turner, a virtuoso painter who belongs to no particular school and whose influences are so broad and vision so intense as to make it almost impossible to say where he comes from.
These two artists, separated by 150 years, can be mapped on to each other in plenty of ways, starting with their motifs. While Turner painted majestic ships, crumbling castles, classical temples and arcadian landscapes, Doig does men in canoes, isolated farmsteads, massive modernist apartment blocks and eerie landscapes. The common denominator is a taste for the sublime, a sense of the smallness and vulnerability of the individual.
Turner's motifs came from the familiar genres of painting of his day, those Dutch 17th-century seascapes, the landscapes of Claude Lorrain, and suchlike; Doig's come from a world that is equally familiar to us, that of Seventies American cinema, the horror films of John Carpenter, John Boorman's Deliverance and, more recently, the Coen brothers' thrillers.
That's where all those solitary figures floating in the middle of lakes are from, and all those secluded expensive homes which we glimpse like intruders approaching through a forest. There's even an American police car at the bank of a lake in one of his paintings.
Then there are the atmospheric effects: the artists don't share the same weather preferences but work with comparable freedom and control.
Where Turner did fogs and sunsets, Doig does autumnal reflections in lakes, falling snow and evening light dappling through a forest. There's a Turner-esque bravado in Doig's brushwork. No false modesty here - in one picture, he shows us how to paint a diaphanous white curtain in front of a window with 10 long strokes of transparent white paint on a broad brush.
Turner, so the famous story goes, went round touching up his paintings with dabs of pure red and white and yellow the night before an exhibition opened, so that they looked brighter than his rivals. Doig's version of this? His mesmerising and intricate painted surfaces dappled with myriad tiny dots of pigment outsparkle and out-glitter those of all his contemporaries.
Historically, Doig slots into the revival of painting which began in the Nineties, flying under the radar of much of the art world in a decade dominated by the popconceptualism of the YBAs and "relational" artists, and by German photographers such as Andreas Gursky (whose panoramic aesthetic often featuring tiny skiers dotting an Alpine vista Doig seems to quote in one of his own paintings, Ski Jacket, 1994).
Yet Doig was not like the other emerging painters of this generation. He didn't make abstract oils based on deliberate ugly "mistakes", in the way that Albert Oehlen did. Neither did he debunk the myth of the genius oil painter by employing a purposefully wan and amateurish style, like Luc Tuymans.
Equally, he avoided the expressionism of Marlene Dumas and her predilection for working like a documentary photographer, in series and on social themes. He wanted his paintings to be beautiful, like Elizabeth Peyton's.
But Peyton made soppy teenage oil sketches of pop stars and celebs, ironically and self-consciously quoting an archetypal, pretty style as a form of self-defence. Doig has never done that. There was, in short, nothing apologetic about Doig's painting. Rather, he returned to that moment, at the beginning of the 20th century, on the eve of abstraction when painting ruled the world, its proud purpose to be beautiful.
Doig's paintings share, at different moments, Monet's and Pissarro's love of reflections, the magical stained-glass palette of early Kandinsky, the soft dreamlike textures of the gouaches of the symbolist Odilon Redon, and Gauguin's trick of using large patches of landscape for expeditions into abstract colour.
Born in Edinburgh in 1959, Doig lived in Canada until he was 20, and returned there at the end of the Eighties for a three-year stay. The Canadian connection is important because of another influence on Doig, the Group of Seven, Canadian post-impressionists who deserve their own show in London. Credit where credit is due, Paul, from the Tate Britain's cloakroom, reminded me of this obscure school.
Here in the work of Tom Thomson, A Y Jackson and Lauren Harris one can find the autumnal maple leaf palette, the golden sunlit highlights and falling snow and the blueness of twilight along the Lakes which must have inspired Doig.
It's all there, yet Doig is not one of those lesser new painters who cook up a soup "quoting modernism". He's got his own style - as you can see in a picture such as Reflection (What does your soul look like), 1996.
Thin washes of colour divide up the canvas into shoreline and lake, and then the marvellous surface of splodges, droplets and the tracery of lines builds up the reflections and floating leaves, all in a harmonious palette of yellow, ruddy brown and white. Finally the rule-breaker: the blue of the cropped trousered legs of a man, a dissonant human presence.
In the early paintings one can see the primacy of the abstract surface: in Baked (1990) a seascape emerges, almost only by virtue of the silhouette of a boat, from a texture of horizontal crimson brushstrokes, like the grain in a old piece of wood, and a grid of white and yellow dots.
It may be too neat a formulation, but just as in the works of the early 20th century, one can see modernist painters struggling to evolve abstract compositions from real scenes, with Doig it's the reverse - he is pulling real scenes out of an abstract surface. It's traditional and original. As the saying goes, he's an old dog with new tricks.
Is it too nice? Way too beautiful? Walking-around this exhibition, the thought kept crossing my mind. Some of the paintings, I worried, resembled an overiced child's birthday cake - the spatters of white, yellow, blue and green like hundreds and thousands and the congealed dots of thick oil paint like Smarties.
In Gasthof zur Muldentalsperre (2000-2), two strange figures, the one in a top hat, the other in a 19th-century military uniform, stand at the entrance of a spookily long driveway which winds out around a lake. They look at us as if they are greeting approaching guests. Twilight has set in, and the green of the trees merges into the darkening sky. The fading light catches white flowers in the fields, while overhead stars, in the same hue, are visible.
So far, so Tim Burton Hollyweird fantasy-horror-movie. But Doig has then seized the mottled texture of the stone wall and decided to use each individual stone to present a rainbow of jewel-like colours.
It's a tour-de-force, but also a bit too sweet for my tooth. Those of us with a knowledge of the cheesiest of 20th-century European painting may find ourselves turning the next corner of this exhibition fearful that Doig could go too far down this road, and end up - nightmare of nightmares! - like Friedensreich-Hundertwasser, a colourful, wacky, highly commercial Austrian 20th-century painter, who could best be described as a tie-dye hippy Klimt and whose prints hung in every suburban German home in the Eighties.
It never comes to that. Doig's fairytale paintings are ultimately closer to those Kandinsky executed at the beginning of the last century of Russian peasant folk scenes and knights on horseback than to Hundertwasser's kitsch.
But perhaps Doig has himself been aware of the danger. As the last room of the exhibition shows, he has changed his style in recent years, eschewing decorative surfaces and the sublime, producing instead paintings of graphic simplicity in which the Gauguin influence appears ever stronger, though paradoxically bleached out, as if faded by the passing century.
It's almost as if the artist imagines himself as a latterday Gauguin - since 2002 he has lived in Trinidad, and his recent work has the feel of a tropical idyll. In Figures in Red Boat (2005-7), the artist's cherished canoe and reflections are rendered with the thinnest washes of diluted paint, and their messy gravity-driven dribbles.
The canoe is no longer filled with one solitary figure; instead there's a bunch of people, conversing or sitting in contemplation. Instead of an accretion of detail, there's practically none.
Still, those that are there are telling: Doig has set up a poetic - and complementary colour - contrast between the ghostly red boat and the sharp green fronds of palm trees on the distant shoreline. With its unfinished aesthetic, this is such a radical departure in style that it reminds the gallery-goer that this Doig is an artist who has probably not even reached the halfway point of his career.