The 22-year-old left-back moves to Newcastle on a five-year deal after the two clubs agreed an undisclosed fee.
"Jamal is a very exciting prospect," Bruce, the Newcastle head coach, told the club's official website.
"I watched him flourish in the Championship two years ago and he made a real fist of it in the Premier League last season.
"He is only 22 years old and has a wonderful future ahead of him so I hope he is a Newcastle United full-back for years to come."
Lewis made 28 Premier League appearances for Norwich in 2019/20, scoring once.
Newcastle open their Premier League campaign at West Ham United on Saturday.