The 21-year-old has made the switch following 21 appearances for Welsh last season in the Aviva Premiership.
Tigers director of rugby Richard Cockerill told the club's official website: "Tom is a young up-and-coming prop and he joins a very good bunch of promising young front rows at the club.
"This is a good environment for Tom and the other young guys to learn their trade alongside senior props Marcos Ayerza, Boris Stankovich, Logovi'i Mulipola, Dan Cole and Jerome Schuster."
Bristow added: "I enjoyed my experience with London Welsh in the Aviva Premiership last season but I'm happy to be here now and delighted to have the opportunity with Leicester Tigers. I'm looking forward to settling in during the pre-season and being part of the Tigers squad."
Meanwhile, Leicester will begin the defence of their title against Worcester Warriors on September 8.