(ex) meteorologist here: Any cloud has the ability to generate turbulence as phase changes take out or add heat energy to the surrounding air creating density changes and therefore vertical motion; since it's not uniform across a cloud this generates turbulence. You get more 'bang for your buck' in liquid water clouds as condensation/evaporation is much faster than deposition/sublimation in ice clouds, and the amount of water vapour that air can hold (saturated vapour pressure) is much lower at lower temperatures (12hPa at +10C, 6hPa at 0C, 0.06hPa at -50C). Clouds of any nature will only form where the relative humidity is approx 100% and cirrus clouds can precipitate out (often seen as "mare's tails"), but will more often slowly sublime away.
Now the amount of turbulence associated with cirrus A versus cirrus B has a lot more to do with how it was formed; if it's associated with gentle uplift from a warm front several hundred nm away it might be smooth or turbulent depending where it is positioned relative to the jet stream, whereas if associated with the remnants of a CB it could be smooth or turbulent depending on the local flow and stability. So to the original question - "Do [high altitude] clouds cause turbulence, or vice-versa" the answer is a rather unsatisfying 'it depends....'