As a baby-faced child with a remarkably serious stare, Shahrzad Rafati excelled at math and science, though hardly in the likeliest of settings. One of her first schools was a gender-segregated stable — borrowed from farmers in northern Iran — that cows and roosters wandered into during class. Her family had sought safety there from Tehran, where she’d seen people’s skin burned off by Iraqi dictator Saddam Hussein’s chemical weapons and her neighbors’ houses bombed to oblivion during the eight-year-long Iran-Iraq War.

It was not a life, or a country, that was for her, she realized, at age 13.

Today, Rafati is the founding CEO of one of the fastest-growing online video companies in the world. It’s called BroadbandTV, and it gets 5.7 billion views a month — up a stunning 80 percent in 10 months. You will surely be hearing about this explosive video business and about the 35-year-old entrepreneur who heads one of the biggest players, the biggest in America by some measures. But you might not hear all that much about how she managed to escape Iran as a teen, leaving her family behind to arrive in Vancouver, speaking almost no English.

A striking and statuesque figure with straight raven hair, Rafati has jumped into a business that focuses on a rapidly growing, though slightly behind-the-scenes, area of video called multi-channel networks. These MCNs aggregate video from creators and feed them into channels — music, comedy, cartoons, etc. — through YouTube and, increasingly, Facebook, while helping to drum up advertising and, naturally, taking a cut. The top channels command 100 billion monthly views. “The scale is breathtaking,” says Chris Dorr, executive director at the Global Online Video Association.

Indeed, the business presents an “existential” threat to the much bigger revenue streams of traditional broadcasters, says Richard Broughton, an industry analyst at Ampere Analysis in London. Now, digital vehicles such as the one Rafati has created and grown are being bought up or invested in by big TV broadcasters, including Disney. RTL, the European TV broadcaster owned by German media giant Bertelsmann, invested $36 million in BroadbandTV, a company Broughton estimates could be worth more than $500 million.

Yet big as her company is, Rafati surprises me when I meet her, not quite the stern, commanding personality her publicity photos suggest. Instead, I meet someone with a ready, spontaneous smile and infectious enthusiasm, as she leans forward speaking nearly flawless, rapid-fire English with just a hint of an accent while gesturing with her hands. In New York from Vancouver for business meetings, she wears a jet-black skirt suit over a white blouse while talking a lot about improving herself and making a big impact on the world — in an idealistic tone reminiscent of middle school dreams, absent any hint of self-doubt or irony.

This woman in a hurry founded BroadbandTV a decade ago, after visiting the Consumer Electronics Show in Vegas as a shy intern for TIO Networks, a billing-service provider. But Rafati also had “a quiet confidence about her,” recalls TIO’s CEO Hamed Shahbazi, and Rafati imagined the possibilities of feeding in video from anywhere in the world. So at 25, she decided to go after sales of set-top boxes — with Shahbazi’s backing. It took just three months to realize that selling hardware, basically a commodity product, was a terrible idea on which to build a big company, so she pivoted quickly to her strengths — math and software — while learning an early lesson: “I’m a big fan of quick failures,” Rafati says.

Born into a family of Persian entrepreneurs — her mother founded a textiles company, while her father started a real estate venture — Rafati’s turn to business was perhaps not a surprise. “I grew up in a family of entrepreneurs, and I always felt I wanted to make a big impact,” she says. But after the revolution, “it was a different atmosphere,” and opportunities that they once took advantage of to start and build businesses had disappeared, not least because of widespread poverty and unemployment. The family successfully applied for entry visas to Canada, but only the 17-year-old daughter took the plunge, over her parents’ opposition at that.

In the end, those fears that she could not make it on her own in a foreign country without family proved wrong. After arriving, Rafati stayed with a friend and crammed English while working as a math tutor for Persian students. She compensated for her weak grasp of English by excelling in a computer-science degree before starting her own nonprofit aimed at connecting donors and charities online, a project that sparked an appetite for building things. Shahbazi, her investor and mentor, says it was Rafati’s constant vision that kept him by her side, even after she shifted her young company out of video hardware.

By the time Rafati saw BroadbandTV’s big breakthrough, in 2007, her team had developed algorithms in-house and was starting a trial run for the NBA. (“I’m a big basketball fan,” Rafati says.) They used software to find fan-uploaded YouTube content and linked it to the NBA, allowing the basketball league to earn advertising revenues and embrace its online fans — instead of branding them pirates and making enemies out of them. It’s still a customer today — as are Sony and A&E — and BroadbandTV is expanding across the globe, from Latin America’s Spanish and Portuguese markets to Europe, India and Japan, where it already has offices.

All this growth, of course, may not continue; for one thing, Broughton warns, “Most of these business are not making much money.” (For her part, Rafati says BroadbandTV was profitable when RTL invested in it in 2013 but won’t say if that’s still true today.) But in the meantime, she says she’s working on being the boss — “maybe I’m too direct,” she says. On a recent vacation, she studied transformational leadership at Oxford and has taken care to sleep at least five or six hours a night. But when the holidays come around, she skips the Muslim ones; Rafati says she left religion behind in Iran. Still, the child with all those jumbled memories has not forgotten. “I may want to go back to Iran someday,” she says.

Correction: An earlier version of this article misidentified Shahrzad Rafati’s age in a photo and her venture that closed.