Crichton has been linked with an exit from the club for the best part of 18 months over salary cap space, with the Origin and Kangaroos back rower leaving it late to sign a new deal.
He has been able to negotiate with other clubs since November 1 last year, while a move to rugby union was also floated.
He will remain at the tri-colours until at leas the end of 2024 though, with the star forward currently away with the Kangaroos at the Rugby League World Cup.
“I've made no secret of the love I have for the Sydney Roosters. This is my home and I'm really excited to have re-committed to the Club for the next two years,” Crichton said in a club statement.
“I know I have so much more to give and that my best football is in front of me, and I'm looking forward to continuing to play my role for this great Club."
Turpin has been linked with the club for the last couple of weeks following the conclusion of his time in Brisbane, having fallen behind both Cory Paix and Billy Walters in the pecking order last year.
With Walters confirming a new two-year extension at the Broncos, and young gun Blake Mozer also likely to push for an NRL debut this year, Turpin's time at Red Hill wasn't extended.
The Roosters were on the lookout for a back up however after Sam Verrills was granted a release to join the Gold Coast Titans for 2023, and Freddy Lussick gained a release to link up with the New Zealand Warriors midway through 2022.
“Jaxson is a skillful outside back whose best football is still ahead of him, and Jake is a specialist hooker who will provide some terrific depth to our squad," Anderson said.
“We look forward to welcoming Jaxson and Jake when they officially join the Club at the start of pre-season next month.”