Although the Turing Machine is a theoretical model to describe computing power, in its basic idea today's computers are still (quite sophisticated) implementations of this basic concept, although you have random memory access and many ingenious optimisations to squeeze out as much performance as possible.
Anyhow, the computability is precisely the same as long as we don't hit the memory limit of a computer.
In Star Treks The Next Generation and Voyager we see lots of features and limitations of computer systems. For example, a hologram cannot be stored/copied as we would assume files can be handled in today's computers. The time a computation takes (whether it's on the ship's core or on Data, or whatnot) is also not always consistent with our intuition, which probably accounts for dramatic effect in particular scenes. Whenever "the computer needs time to compile the information" it's really an excuse to give the characters a chance to chat.
So, ultimately, is the core idea of the computers we seen in Star Trek (or more precisely on the NCC-1701-D, if you like) still a descendant of Turing's design, or is there a fundamental difference/innovation. Can these computers compute something a TM can't (e.g. decide the diagonal language on TMs)?