I'm not a historian, but I did study Computer Science (CS). In CS, Alan Turing is known for a lot more than just his WWII work. CS programs, in the US at least, talk a lot about Alan Turing because his work is foundational to the field. He created a very simple mathematical model of a computer called a Turing Machine that can be used to show the power/limits of computers in general. Even without his contribution to the war effort, he'd be very famous in CS circles. If you learn about Isaac Newton when studying intro to Physics, you study Alan Turing when doing intro to CS.
Additionally, he's also in popular culture because of the famous "Turing Test", which is connected to answering the question "can machines think?". Given AI is still an open and important field, Turing's work is still relevant today.
Finally, he's a gay rights icon/martyr because he killed himself after being was persecuted for being gay and forced to take female hormones.
My answer is meant to point out that Alan Turing's importance goes beyond his WWII Enigma cracking work, and I can't speak directly to Turing's importance vs Arne Beurling's.
I'm not sure it's really possible to quantify "better" when comparing the work of mathematician/codebreakers like Beurling and Turing (or others such as Marian Rejewski). Turing certainly made a number of enormously important contributions to the work of Bletchley Park, and with Sweden's neutrality the output of Bletchley Park had a more direct impact on the outcome of World War II; Turing particularly worked on naval Enigma, which was an important factor in the Battle of the Atlantic against German U-boats, of which Churchill famously said "...the only thing that ever frightened me during the war was the U-boat peril." As /u/abatabat says, Turing also had a degree of fame in the fields of mathematics and computing from his pre- and post-war work even before details of his codebreaking work were known (the breaking of Enigma remained a secret until the 1970s) and something of an eventful life. To an extent that can overshadow the work of others in the field, I don't think it's deliberately done for the most part, just that the dense mathematics of machine cryptography is rather hard to convey whereas locating U-boats as a result of breaking codes, or something like the Turing Test, is much easier to grasp.
For example, another British mathematician, Bill Tutte, performed a very similar feat to Beurling in reverse engineering a teleprinter cipher machine based only on intercepted messages (Beurling the Siemens T52, Tutte the Lorenz SZ known as "Tunny" at Bletchley Park), and though Tutte's achievements are mentioned in a lot of books about Bletchley and were covered in a BBC documentary largely dedicated to him and Tommy Flowers (an engineer who built the first programmable electronic computer to crack Tunny) his name is still barely known (although probably slightly more than Beurling, who I confess I wasn't aware of until now).
For further reading on Bletchley I can recommend Codebreakers: The Inside Story of Bletchley Park (edited by Hinsley & Stipp), and Action This Day (edited by Smith & Erskine)