Steam enters at the left from an opening on the upper half of the casing, flows along the shaft axis (hence the term axial-flow turbine), then the steam exits the turbine from a large opening at the right, on the upper half of the casing. This is the typical flow path on all modern steam turbines.
However, the Parsons Turbine that powered the ship Turbinia was a Radial-Flow turbine. In this turbine, the steam flow path was outward from the shaft to the casing. Very fascinating!
Below is a cut-away of a Parsons Turbine that was once in a Dreadnought (an early Battleship).
To be clear, the steam in this turbine also travels down along the turbine shaft, but it is only making power while traveling outward (radially) toward the casing. Steam enters at the lower left in this picture. It then travels outward from the shaft and passes through 4 turbine stages mounted on a disk, before being routed back to the center. The process repeats on the next disk.
From what we can see, it appears there were six disks (for high pressure steam) with four stages each, and what appears to be two larger disks (Low pressure steam) with seven stages.
Below is a closer look at the main steam inlet and first few stages of this interesting turbine. As an aside, to the left is the shaft seal that prevents the steam from leaking out.
Having been through a few steam turbine overhauls, it's easy to see why this particular design is now mainly a historical curiousity. Each set of stationary blades appears to be attached to a casing segment. You can see where the segments fit together where there are lines in the red paint on the upper photo.
There are a couple of large bolts that appear to be holding all the casing segments together in the upper photo.
From a maintenance standpoint, this machine would be pretty difficult to take apart and repair. Because the stationary blades seem to be attached directly to the turbine casing, any stationary blade damage would require that a segment of the turbine casing be removed for repair.
Operationally, there were probably steam leaks between casing segments, and alignment issues between segments as well. To compensate for the alignment, I would expect that clearances were probably quite loose, resulting in loss of efficiency.
Nevertheless, because it was such a huge improvement over piston steam engines, this was a successful turbine in its day, and was installed in a large number of warships in the early 20th century.