Mr. Shapiro, conservative thinker, entertainer, trash talker and destroyer of weak arguments, has been called the voice of the conservative millennial movement. He represents the tastes of an emerging political class: If Rush Limbaugh is someone your dad listens to on his car radio, Mr. Shapiro, 33, a graduate of Harvard Law School, is the cool kid’s philosopher, dissecting arguments with a lawyer’s skill and references to Aristotle. He exists in places that young people inhabit — podcasts, Facebook, Twitter and YouTube. His podcast, The Ben Shapiro Show, is downloaded 10 million times every month. Seventy percent of his audience is under the age of 40.

“Ben is the source of a lot of philosophy for young conservatives,” said Kassy Dillon, a senior at Mount Holyoke College in Massachusetts, who has written for Mr. Shapiro’s website, The Daily Wire. “I think it would be really hard to find a young conservative who doesn’t know who Ben Shapiro is.”

Mr. Shapiro is trying to define conservatism at a time when its meaning is up for grabs and the Republican Party, and traditional conservative media outlets like Fox, are built for older audiences.