Ms. Wojcicki was working for Intel at the time, and one day, she wasn’t able to find an important piece of information because Google was down. She realized that she had become dependent on “the site developed by those two dudes in my garage.” Ms. Wojcicki joined Google as its 16th employee and its first marketing manager.

With a baby on the way, she was going to work for a company with no revenue. She helped turn Google into a juggernaut, developing its signature advertising product, AdWords, and overseeing its first foray into content sharing, called Google Video. In 2006, swallowing her pride, she encouraged Google to acquire a rival service called YouTube for $1.65 billion. (Morgan Stanley recently estimated that the division is now worth $160 billion.)

Over the years, Ms. Wojcicki built a personal fortune estimated in the hundreds of millions, and she remains a close confidant of Mr. Page, who is now the chief executive of Google’s parent, Alphabet. One former executive told me, “Susan is the first cousin of the Google royal family.”

Ms. Wojcicki’s biological family is remarkably accomplished. She is the oldest of three sisters; the middle one, Janet, is an anthropologist, epidemiologist and former Fulbright scholar; and the youngest, Anne, co-founded 23andMe, the DNA testing company. (Anne was also married to Mr. Brin until 2015.)

Being part of a high-profile clan can cause confusion. Chris Dale, a YouTube spokesman, once learned that People magazine planned to run a story that Susan Wojcicki was dating Alex Rodriguez, the former New York Yankees slugger. Mr. Dale recalls sheepishly calling his boss — who had been married for more than 20 years — in the middle of a parent-teacher meeting to ask if she was having an affair. She was not. Anne, not Susan, had been seeing Mr. Rodriguez.

Since taking over YouTube, fame has been something Ms. Wojcicki has tried to avoid. But she is warm and approachable in person, and for one recent interview, she arrived at the office dressed like any other Google employee, with black cowboy boots and a backpack slung over one shoulder, ready to discuss her track record. “One way I think about some of the decisions is putting myself in the future and thinking: in five or 10 years, what will they say?” Ms. Wojcicki said. “If someone were to look back on the decisions that we’re making, would they feel we were on the right side of history? Would I feel proud? Will my children feel like I made good decisions?”

Whether vacuous or violent, YouTube troubles arise at a dumbfounding pace. Last year, YouTube banned both the “Tide Pod” challenge, which involved biting into laundry detergent pods, and “No lackin” videos, in which participants aim guns at each other. Confronted by one crisis after another, Ms. Wojcicki has introduced policies to curb the spread of problematic videos and raised the bar on which videos can carry advertising. The challenge is cleaning up the mess while keeping her various constituencies — users, creators, advertisers, parents, lawmakers — happy, or at least not irate.