This seems to be true in the context of (some areas of) computer science but not generally.
One reasons has to do with the Church's Thesis. The main reason is that some experts like Godel didn't think that the arguments that previous/other models of computation capture exactly the intuitive concept of computation were convincing. There are various arguments, Church had some, but they did not convince Godel. On the other hand Turing's analysis was convincing for Godel so it was accepted as the model for effective computation. The equivalences between different models is proven later (I think by Kleene).
The second reason is technical and a later development related to the study of complexity theory. Defining the complexity measures like time, space, and nondeterminism seems to be easier using Turing machines than other models like $\lambda$-calculus and $\mu$-recursive functions.
On the other hand, $\mu$-recursive functions were and are still used as the main way of defining computability in logic and computability theory books. They are easier to work with when one only cares about effectiveness and not about complexity. Kleene's book "Metamathematics" was very influential for this development. Also $\lambda$-calculus seems to be more common in CMU/European style computer science like programming languages and type theory. Some authors prefer the RAM and Register Machine models.
(It seems to me that for some reason Americans adopted Turing's semantic model and Europeans adopted Church's syntactic model, Chruch was American and Turing was British. This a personal opinion/observation and others have a different view. Also see these papers by Viggo Stoltenberg-Hansen and John V. Tucker I,II.)
Some resources for further reading:
Robert I. Soare has a number of articles on the history of these developments, I personally like the one in the Handbook of Computability Theory. you can find more by checking the references in that paper.
Another good resource is Neil Immerman's computability article on SEP, see also Church-Turing Thesis article by B. Jack Copeland.
Godel's collected works contains lots of information on his views. Specially introductions to his articles are extremely well-written.
Kleene's "Metamathematics" is a very nice book.
Finally, if you are not still satisfied check the archives of the FOM mailing list, and if you cannot find an answer in the archive post a an email to the mailing list.