2013 - Gravity
In 2013, Alfonso Cuarón's visual masterpiece Gravity was nearly a shoe-in for Best Picture. Raking in seven Oscars (it was nominated for ten), it ultimately lost the Best Picture race against 12 Years a Slave. Still, that's not to say that it wouldn't have fared well in a popular vote.
Some of the most popular movies of 2013 included The Hunger Games: Catching Fire, Frozen, Iron Man 3, and The Great Gatsby. That said, Gravity would no doubt have edged them out for a win given the Oscar love it had already been flooded with. Really, its biggest competition - assuming this category allows animated movies, which may turn out to be an exception - was Disney's Frozen. It epitomized what makes a popular movie (making a lot of money and being impossible to ignore), but Gravity was already an Oscar darling, and if nothing else, this win would have doubled as a sympathy vote from Academy members for denying it the Best Picture win.
2014 - Interstellar
From the Academy's perspective, Christopher Nolan isn't quite worthy enough of an Oscar just yet. He's mastered the superhero genre, the war picture genre, and the science fiction genre, but they're still waiting on what they must presume will be his masterpiece. That said, from a "popular vote" perceptive, Christopher Nolan may well have earned himself a second win in the very same category for his time-and-space opera Interstellar had the "popular" category existed back in 2014.
The most well-received and popular movies that also stood a shot in this category included Guardians of the Galaxy, Dawn of the Planet of the Apes, The Hunger Games: Mockingjay - Part 1, and maybe even something like The Fault in Our Stars, to recognize mainstream movies that weren't necessarily massive blockbusters. Still, though, Interstellar would have likely earned the Academy vote, considering the pedigree of talent involved, as well as the fact that it had already earned a slew of nominations (as well as a single win for Visual Effects).
2015 - Star Wars: The Force Awakens
Hollywood loves a good comeback story. And while that might extend to the Academy to some degree, they're a bit pickier with the sort of stories they officially recognize. So, when the Star Wars franchise essentially rebooted itself following George Lucas' mostly underwhelming prequels, the Academy was generous with J.J. Abrams' Star Wars: The Force Awakens. The movie earned five nominations, but in technical categories (save for John Williams' score), with very little chance at edging its way into the Best Picture category. That said, had its been considered on its popular merits, The Force Awakens would have undoubtedly taken the crown.
To be fair to its competition, there's a chance that movies like Mad Max: Fury Road (which was already a Best Picture nominee and surprise favorite), Creed, or Paddington could have swept in as a favorite, but given Star Wars' longstanding history with the Academy, as well as the fact that it was 2015's box office champion (and, yes, how it's now spawning its own theme park), it's kind of impossible to beat Star Wars in a popularity contest.