Oliver Jeffers' right arm is all strapped up and bandaged: he's recovering from surgery for a torn ligament. "I'm telling everyone an entirely different story," he says. "Some people I tell it was a skateboarding accident. Other people I tell I was mauled by a lion. Or I was in a sword fight. A parachute jump. Skiing. Self-defence. Or I'm a cyborg. You know what they say: never let the truth get in the way of a good story."

Whatever the truth, Jeffers' injury has momentarily slowed him down and he's back in Belfast catching up with family and friends for a couple of days, fresh from the Bologna book fair, where he was celebrating 10 years since the publication of his first book, How to Catch a Star, with a couple of dozen of his publishers from around the world. Before that, he was at the TED conference in Vancouver, hanging out with tech guru Nicholas Negroponte, then he returned briefly to Brooklyn before heading off to Brazil, then on to the Barcelona art festival and then the LA Times book festival. There are also his collaborations with U2 to be getting on with; and his work with Irish novelist John Boyne (author of The Boy in the Striped Pyjamas); and with Eoin Colfer (of the Artemis Fowl series); and his continuing promotional work for The Day the Crayons Quit, his New York Times No 1 bestseller, written with Drew Daywalt; and his major exhibition of paintings coming up later this year; and the publication of his next book, Once Upon an Alphabet, and Eric Schmidt wants him to come and talk about creativity at Google ...

"Aren't you tired?" I ask. He looks a bit tired, in a ruffled rock-star kind of way. "It does feel like I've been running full steam for the past year," he says, sipping hot tea in the piano lounge of Belfast's Europa Hotel. "So it's good to be taking it easy for a while." He talks – in paragraphs, intensely, seriously, spontaneously and eloquently, in his Northern Irish accent – for 45 minutes, with barely a prompting or a question. If this is Jeffers taking it easy, one can only imagine what he's like when he's in full flow.

Jeffers – in case you have somehow missed his extraordinary ascendancy – is a picture-book writer and illustrator, born in Australia in 1977, raised in Belfast, and apparently on the verge of pan-global, cross-media superstardom. His books are rarely reviewed yet his work sells millions of copies, and his images are familiar to anyone who has watched TV ads for Kinder eggs, or been to Starbucks, where he has created in-store displays, or drunk Lavazza coffee, for which he has also done illustrations.

How would he describe himself? "I really just think of myself as an artist and storyteller," he says. "That's it. That's what I do." He doesn't refer to his books as children's books: they're picture books. And like Maurice Sendak before him – "You cannot write for children. They're too complicated. You can only write books that are of interest to them" – he not only refuses to accept the demarcation between children's books and adult books, but also between books of any kind. "My books are just books," he says. "That happen to have pictures in them. I don't like to think about how my books work or who they're for. All I'm trying to do is tell a story as clearly and simply as possible."

I suggest that the stories he tells – from How to Catch a Star (2004) to It Wasn't Me (2013) – all seem to share a common theme: something to do with loss and grief, and a sense of the tragic in human existence. "Maybe," he says. "But I think you could say the same about most books, and most art." (He's an omnivorous reader, with four or five books on the go at any one time: fiction, non-fiction, everything.) He doesn't think in terms of themes or objectives in his own work. "I don't really write books for people to study and analyse – I write them for people to enjoy. And they'll only enjoy them if I enjoy them. For me, it's fundamentally a self-fulfilling endeavour. When you get it right, it's just so satisfying – it makes you want to do more and to share it with other people. If my work is successful in any way, it's because I'm working things out for myself, satisfying my own curiosity."

A book published a couple of years ago, Neither Here Nor There: The Art of Oliver Jeffers, the first proper catalogue and collection of his work, edited by Richard Seabrooke, reveals the full scope of Jeffers' extraordinary curiosity. Neither Here Nor There showcases Jeffers' fine art work – paintings, collages and installations – much of which explores the relationships between art and mathematics, art and science, and above all, between art and philosophy. Jeffers is clearly not a conceptual artist, but he likes playing around with concepts, perhaps most notably in his recent series of what he calls "dipped portraits", beautiful paintings in oil on canvas, complete and carefully executed, which he then dips in enamel, obscuring the image underneath. What's that all about? Is it about death? There do seem to be ghostly presences throughout his work – people, places and things either not there or out of place. "I just want to make people think," he says. "I want to make them think about what's there and what's not there, what's lost and what can be found." He's considering turning the dipped portraits into some kind of live show later this year. "No one will be allowed to bring their phones, or cameras or watches, and I'll paint the portrait there and dip it in enamel in a specially constructed vat, erasing it so that they'll only have their memories to know what it looked like."

There's little chance of forgetting what a work by Jeffers looks like. His books and his paintings are distinguished by his unmistakable hand-drawn typography, with its wavering lines and eccentric use of capitals, and by his hot-and-cold colour palette, contrasting misty cool blues and greens with vivid foregrounds. His characters are often stripped to their absolute essentials: the boy, the penguin, the moose, with circles and blobs for bodies and heads, and thin straight lines for legs. Reading his books is like wandering into a subtitled child's dreamworld. Lost and Found (2005) was made into a successful TV animation. The Incredible Book Eating Boy (2006) is in development as a live-action drama with Universal Pictures, and with his friend and oft‑time collaborator, the artist Mac Premo, Jeffers last year directed the video for U2's single "Ordinary Love". "The U2 thing just came together by chance," he says. Bono's wife used to read Jeffers' books to her children, and one day he and Bono met for a drink. "And one thing led to another. That's the way I work. Nothing's really planned – it's organic."

In Jeffers' self-defined three categories of work – "the books I make, the paintings I do, and Other" – the U2 video counts as Other. For a long time, he supported his own art by taking on commercial illustration work, but he's now in the fortunate position of being able to pick and choose what Other stuff he wants to do. "I make art for love rather than money," he says. "The difference is, now I'm able to make money by making art for love." After years of exhausting effort – he suffered for a time from insomnia – he's also now worked out a fine work‑life balance. His wife acts as his manager. "There's no way I could manage this amount of work without help. I'm basically running two or three careers at once."

These days, all his multiple careers take place in New York, Jeffers and his wife having left Belfast seven years ago – they received their green cards this past St Patrick's Day. "Of course I miss Northern Ireland," he says. "This will always be home. But Brooklyn has treated me pretty well. It's just a wildly inspirational place, filled with interesting people who want to accomplish and do interesting things." A few of those interesting people share studio space with him at the Invisible Dog Art Center, one of those places that seems to have sprung fully formed from the head of a hipster: an art space that was formerly an invisible dog-lead factory. "You couldn't make it up, could you?" admits Jeffers. And I'm not sure if he is making it up – he has lost none of that wry, wild Belfast wit.

What he is currently making up – apart from implausible explanations for his hand injury – is Once Upon an Alphabet, due for publication later this year. Comprising 26 four-page interlinked stories, it is unlike any of his previous work. "To be honest, this is just a book of pure silliness," he says. "It's a book of joy. A kind of culmination. It's a collection of sketches and ideas that I've been working with for some time which I've somehow shaped together into a book. I didn't know what to do with them all and then they all started to come together in my head."