Peter Hildebrand choked back tears as he told the crowd about his daughter, Daisy. She was 8 years old when she died in April, one of the two unvaccinated children lost in the measles outbreak that tore through West Texas. “She was very loving,” he told the audience.

It was Day 2 of the annual conference of Children’s Health Defense, the organization of vaccine critics previously led by Robert F. Kennedy Jr., who is now the U.S. health secretary. Mr. Hildebrand had been asked to speak on a panel titled “Breaking the Mainstream Media Measles Narrative” at the conference, which brought 1,000 people to an event center in Austin, Texas, this month.

Mr. Hildebrand spoke about mistrusting Daisy’s hospital doctor, who he said talked to his wife about measles when he was out of the room. “You kn

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