The Northern Ireland fullback, 22, is highly rated and was targeted by Liverpool before they signed Kostas Tsimikas.
He has now arrived at St. James' Park after the two clubs reportedly agreed a £13.5m.
"Highly-rated full-back Jamal Lewis has become Newcastle United's fifth signing of the summer after joining the Magpies from Norwich City," the Magpies signalled in a statement published on the club's official website.
"The 22-year-old Northern Ireland international has signed a five-year contract at St. James' Park after earning rave reviews for his performances last season despite the Canaries' relegation from the Premier League.
"Lewis, who started his youth career with his hometown club, Luton, before joining Norwich's Academy in 2014, won a Sky Bet Championship title-winners' medal with the East Anglian side in 2018/19 and was also named in the league's Team of the Season before further enhancing his reputation in the top flight last term.