Thompson, 24, joined Norwich from Swindon in September 2014, but Achilles injuries have limited him to 14 games.
Shrews boss Sam Ricketts said: "Louie's a great young player.
"Great player, great pedigree and great person. He's really dynamic."
Sean Goss from Queens Park Rangers has also been signed on a two-year deal
Shrews play Rochdale in League One on Saturday.