The 25-year-old joined Scarlets in 2007, having previously played for Carmarthen Quins and Llanelli, and has made a total of 119 competitive appearances.
Turnbull has represented Wales at U18, U19 and U20 levels and participated in two IRB Under-19 World Championships.
The Haverfordwest-born flanker was named in the Wales senior squad for the first time for the 2011 Six Nations and made his debut as a replacement in the 24-6 win over Scotland at Murrayfield. He was also selected for the 2012 tour to Australia.
Director of Rugby Phil Davies said: "We are really pleased Josh has agreed to join the Blues, it's a big and important signing for us.
"We were looking for an experienced player and an out and out number six. With Andries Pretorius set to leave us and Michael Paterson having gone last year, it was an area we were focused on.
"The opportunity to get Josh was massive for us. I know him really well from my time at Scarlets, as I gave him his debut, and he is a player I've always followed and admired.
"It's exciting for us to bring in such a powerful player with that pedigree and quality. Josh adds balance and depth to our back row resources and his experience will be invaluable to the younger players who will learn so much from him."
Turnbull said: "When I spoke to Phil Davies I was excited about what he had to say about the future. I have been at Scarlets for seven years and feel now was the right time for a new challenge.
"Blues are looking to add depth at number six and that is where I want to play. I have played across the back row, but I want to focus on one position and move my career up another level."
Turnbull added: "I owe a huge thanks to everyone at Scarlets. It was actually Phil Davies who gave me my first professional contract, I will always be grateful for that. Since then I've worked with some great coaches at players and I will leave with many happy memories.
"I want to finish on a high with Scarlets now and climb the RaboDirect PRO12 before I make the summer switch to Blues.
"Ultimately I have ambitions to play for Wales again. I haven't done that since the autumn of 2012 and that is a huge target for me.
"I feel focusing on one position week in and week out will improve me. Hopefully I can move my game up and push my international claims once again."