There are two competing effects.
If you consider entry to the kind of "design gust" called for by design standards, then you are considering a rapidly changing vertical component of aircraft velocity with respect to the air mass. if for now we assume the aircraft velocity with respect the the ground to be constant, we get a rapid change in angle of attack, and hence lift. Since a given amount of lift will accelerate a lighter aircraft more, it follows that the lighter aircraft sees more "g" and thus the components also see more accelerations. Since the components are of fixed weight, it follows that their fittings see higher loads at higher "g" - as mentioned by ft.
But consider those components which are varying with weight, and those fittings which take loads associated with the aircraft mass, not component masses - things like wing-to-fuse fittings. In this case, although the "g" has gone up the associated weight is lower, and these cancel. Which would tend to indicate that weight is not a factor in loading of major airframe components in turbulence.
but
We made an assumption that "the aircraft velocity with respect the the ground [will] be constant". The reality is that this is not the case. Gusts aren't instantaneous, they take time to build to peak. So the aircraft has time to respond a bit to the gust, which tends to reduce the effect of the peak gust disturbance. Heavier aircraft, however, possess more inertia, and so respond slower, relieving less of the load. Thus the actual peak load on a heavy aircraft is higher, and thus the airframe level loads are actually higher than for a light aircraft, although the peak 'g' is still less.
Therefore, the effect of turbulence is:
* better ride quality for havier aircraft (lower peak 'g')
* less component loading for a heavy aircraft (lower peak 'g')
* worse airframe loads for a heavy aircraft ('g' times mass is higher)
With all the usual caveats about specific aircraft characteristics, of course ....