Norman returned one of the Panthers' four interceptions of rookie Jameis Winston for a touchdown, and Newton tossed a pair of TD passes to Ted Ginn, Jr. to key a 37-23 victory that kept Carolina unbeaten Sunday.
Winston threw for a career-high 287 yards and two touchdowns. But Norman picked off the young quarterback's first pass of the day and returned it 46 yards for an early 10-0 lead, and the Panthers wound up forcing five turnovers overall.
Newton threw scoring passes of 6 and 12 yards to Ginn. He had a modest day, statistically, with 124 yards passing and 51 more rushing, but the fifth-year pro didn't throw any interceptions and had just one turnover on a day when play was affected at times by heavy rain.
A week after making a game-saving interception in the end zone against New Orleans, Norman also returned a second interception 34 yards to set up a Carolina TD.
According to the NFL, the cornerback became the first Carolina player to return a pair of interception for TDs in the first four weeks of the season. He's the first to do it in the league in the past five years.
The Panthers improved to 4-0 for the first time since 2003, when they began with five straight victories and went on to appear in the Super Bowl.
The Bucs (1-3), have lost five straight in the NFC South rivalry and have dropped 11 consecutive homes games -- 10 under second-year coach Lovie Smith -- dating to December 2013.
Winston, the No. 1 overall pick in the draft, completed 26 of 43 passes and was sacked twice. He threw TD passes of 10 yards to Charles Sims and 7 yards to Vincent Jackson, who had 10 receptions for 147 yards.
Doug Martin rushed for 106 yards and one TD for the Bucs, who outgained the Panthers 411 yards to 244.
Rookie Kyle Brindza had another tough day kicking for Tampa Bay. He missed three field goals and an extra point, during last week's 10-point loss at Houston and missed two more field goals and an extra point against Carolina.
It was the second time in four weeks that Winston's first pass of the day was picked off and returned for a touchdown. A fumbled snap on the second play of the game set up an early Carolina field goal, and Winston's second interception -- on a tipped ball -- led to Newton's 6-yard TD throw to Ginn for a 17-3 Carolina lead.
One of the young quarterback's strongest assets during the opening month of the season has been resilience.
The Bucs drove 80 yards in 13 plays to trim their deficit to 17-10 on Martin's 5-yard TD run. And, Winston moved them into position to pull even closer by halftime, but Brindza's 29-yard field goal attempt hit the right upright and bounced away.
The kicker misfired again in the opening minutes of the third quarter, slipping on the wet turf on a 43-yard try that he pulled left after Newton's fumble on the first play of the half was recovered by Bucs cornerback Tim Jennings at the Carolina 25.
The Panthers took over at their 33 and took control for good on the next play, with Newton handing off to Jonathan Stewart for an 8-yard gain that wound up being much more when Tampa Bay's Chris Conte forced a fumble to Dickson snatched out of midair before continuing untouched to the end zone to make it 24-10.