Not sure where you get this from, AIUI an inversion does not
cause turbulence, and above an inversion you will generally find smooth air, you may be thinking of an turbulence inversion but here the inversion is caused
by the turbulence rather than the other way round.
A turbulence inversion often forms when quiescent air overlies turbulent air. Within the turbulent layer, vertical mixing carries heat downward and cools the upper part of the layer. The unmixed air above is not cooled and eventually is warmer than the air below; an inversion then exists.