The drag equation can be used to calculate drag force for both laminar and turbulent flow, if we allow $C$ to vary with velocity in a convenient way. This reinterpretation of the equation and meaning of $C$ is sometimes done in practice, especially if one is measuring the drag and expressing the results of measurements as function $C(v)$. This just a practical matter which is useful in the turbulent regime - physically there is no good reason to use single formula for all regimes.
If we don't allow $C$ to vary with velocity and fix it to a single number, the equation can give drag force accurately only in the turbulent regime, and only for limited range of velocities. When studying wide range of velocities, it is not that accurate. The drag force function of velocity is not exactly a single power. That is why function $C(v)$ is introduced, so that the drag equation can be accurate for a wide range of velocities.
For laminar flow, the expression with constant $C$ is completely wrong and should not be used. The accurate expression is closer to linear function:
F = -kv.
(In fact the drag force depends also on the acceleration of the body, but this can be handled by redefining the mass of the body).
In common conditions in air, the drag force is linear function of velocity only if the Reynolds number Re is much less than 1. Stuff like microscopic water droplets in fog falling down. Things like falling rocks or flying planes in Earth's atmosphere have turbulent flow. The velocity and the body is just too big to allow for laminar flow.