"If you can score on the par-fives, you can shoot a good score and that's basically where I made all my numbers up today," Fleetwood told Golf Channel after making four birdies and two bogeys in addition to his eagles.
"It was just nice to have a day like that and just constantly feel at ease really with hitting good golf shots.”
Bradley, who started the day in second place behind Spanish overnight leader Rafa Cabrera Bello, turned in six birdies to go with a pair of bogeys.
He got a big lift at the 15th hole when he sank a monster birdie putt from 33 feet.
"I don't normally leave the flag in," Bradley told PGA Tour Radio. "I just decided to leave it in out of nowhere, and sure enough it hit the flag right in the middle and went in."
Six players were tied for third at five-under including last week's Honda Classic winner Keith Mitchell (68).
Billy Horschel (71), Venezuelan Jhonattan Vegas (70), Italian Francesco Molinari (70), American Kevin Kisner (69) and Canadian Roger Sloan (69) also were at 139.
Another group of eight were a further stroke back. The group included Cabrera Bello, who skied to a 75, 10 strokes higher than his opening round.
Defending champion Rory McIlroy was seven strokes off the pace after a two-under 70 which included four birdies, three bogeys and an eagle at the par-five sixth.
World number two Justin Rose hit a 70 to sit six shots off the pace but the tournament ended early for five times major winner Phil Mickelson, who failed to make the cut after a six-over 78 that included three bogeys and two doubles.
The frustrated American, who hit an opening round 68, said he would play a practice round on Tuesday before deciding whether to compete in next week's Players Championship.
"I mean, I want to play it, so I would most likely," Mickelson said. "But if I hit it like this, it's pointless, so I got to figure something out."
South African Ernie Els (75) and world number three Brooks Koepka (73) also missed the cut.
(Reporting by Gene Cherry in Raleigh, North Carolina, editing by Nick Mulvenney)
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