The 22-year-old has penned a two-year contract with the League One club for an undisclosed fee.
"Sonny is a commanding defender and someone I have been targeting for some time," Crawley boss John Gregory told the club's official website.
"We have good depth in our central defensive positions now, but Sonny will be challenging the others for a regular place. There was a lot of interest in him. I'm delighted to have signed him."
Bradley leaves Portsmouth after spending just over a year at Fratton Park.