There's no shortage of established Trek technologies that could be applied. Starfleet has mastered small-scale manipulation of gravity, and could be propelling the lifts that way (in addition to providing each with its own localized gravity field and inertia-damping system that makes the movement unnoticeable to the occupants, a must for a system that runs fast through all sorts of straight angles!). Or then the pods could be flying through the network of shafts using any one of the spacecraft propulsion systems shown or implied. Basic maglev (or even mag without lev, with wheels and rails and all) is certainly a "real-tech" option for those preferring such things, but might be inferior to easily available imaginary technologies.
The one thing that doesn't work too well is the "tube mail" concept of vacuum tubes and suction/pressure propulsion, mentioned in the early writers' manual and related works but in contradiction with the non-airtight, air-filled shafts we see.
Why these would be "turbo" lifts might not really need an explanation. Rather than be related to turbines in any sense, they might be deriving their colloquial name from the fact that they are fast, such as turbocharged internal combustion engine vehicles in the primitive 20th century. Or then annoyingly noisy, unlike any post-20th century vehicles. But we could also speculate upon a futuristic operating principle that is ever-so-remotely related to that of turbines.
How do these things change directions? Probably simply by applying their basic propulsion system directionally. Whether the cab itself does most of the work, or the propulsion is mostly built into the shafts, we don't know, although the latter would sound like the more expensive proposition for the extensive shaft systems of big ships or stations.
A maglev or linear-motor-with-rails cab today would be an attractive proposal, as it would eliminate the old-fashioned cables that create a double obstacle for skyscraper construction. There'd be no extra weight for an extra level added, making taller buildings easier to build, and several lifts could operate within the same shaft (up to dozens or even hundreds in an "up"/"down" pair of shafts with sidings), allowing for narrower buildings (i.e. not several city blocks wide at the stem to handle all the necessary shafts!) and more personal service. However, linear motors consume a lot of electricity, and recovering it by braking descending lifts might not be efficient. There have been some heavy-lifting linear engine lifts (such as projected for modern aircraft carriers), but power consumption is highly proportional to the load lifted, unlike with cable systems that negate load issues with counterweights.
Timo Saloniemi