Definitely possible
Yes, buy another (i.e. two or more) Turing Pi and chain them together (via the Ethernet ports).
The onboard Ethernet is just a two port Ethernet switch. Consider it as such, and use accordingly. That is to say, that you can connect one of the Ethernet ports from one Turing Pi to one of the Ethernet ports on the other Turing Pi - so as to daisy-chain them together.
The fact that the ports are, by default, bridged, is of no real concern to you nor other external devices - the ports can still be used as if they were two Ethernet switch ports.
For further information about the bridged nature of the ports, read Trying to understand why Turing Pi 2’s network jacks are bridged.
The Ethernet switch on the Turing Pi is a managed switch so you can reconfigure it, if you really need to.
Alternatives
As an alternative to consider, the Super 6C, can take 6 CMs and you don't need the annoying adapter plates that the Turing Pi requires. Also, each "bay" on the Super 6C has access to its own M.2 slot, which the Turing Pi does not. On the Turing Pi, only one slot has access to the single M.2 slot.
On the other hand, the Ethernet switch on the Turing Pi is slightly more configurable than that on the Super 6C.
In both cases, the Ethernet is just two port Ethernet switches, on each board. Consider them as such, and use accordingly.
You could add just one extra CM to each of the above, via the use of a CM IO Board, and connect it via the second Ethernet port - thus still avoiding the need for an additional network switch.
You could also consider a CloverPi, which takes full-size Pi devices, rather than CMs. It also has a built in Ethernet switch.