A -2.5G event "pulls" you upward 1.5x harder than falling and you absolutely can die from falling down due to head injury - it happens all the time. In reality it doesn't pull, your inertia simply wants to keep you moving as you were so the top of the fuselage "falls" faster than gravity until it connects with your body and is able to transmit that force to counter your inertia. A better way to transmit that force is a seat belt.
This is exactly the same situation as a car hitting a wall and why you need a seat belt in that scenario too. You and the car are moving forward. The car decelerates. When it does so faster than the friction of you in the seat your inertia carries your body forward. The faster the car stops the more energy you have to impact the windshield/steering wheel/etc. The seat belt functions to transmit the force between your body and the car.
Granted they were technically falling, too, but at a slower rate because the force that pushed the plane down - turbulence - acted directly on the plane, but the plane shielded the passengers from the direct effect of the turbulence. Thus the air did not push them down at the same rate that it pushed the plane down onto them.
EDIT: to clarify, my initial doubt/question was about what actually is the cause for the force that, during a turbulence, makes the plane fall at >1G and thus making you hit the ceiling if you are not tied to the seat?
Entropy causing gasses to mix evenly is true... on a large enough timescale.
In reality any gas or fluid will form boundary layers between masses of the substance having different temperatures, moving at different speeds, or having other properties (eg salinity for a liquid).
When flying you can end up passing through these discontinuities - regions where the air is very turbulent on both sides of the boundary. You can also develop different amounts of lift in one air mass vs another due to density. The entire air mass itself may be moving in a different direction and/or at a different velocity on one side of the boundary vs the other.
In a small airplane the effect is more pronounced. On a bright sunny day near the ground I get negative G when passing over a small bit of water followed by positive G when passing over a hot parking lot. It is very obvious where there is a region of hotter less dense air rising vs cooler more humid air.
