Why can transverse waves not be produced in gases?

Transverse waves cannot propagate in gases because there are no forces to cause motion perpendicular to the direction of the wave.

A transverse wave propagates in a solid because each atom of the solid has an "equilibrium location" where the forces on it from all its neighboring atoms balance.

If one atom is pulled out of its equilibrium location by a disturbance, it will tend to be pulled back to that location.

However, when it's out of its equilibrium location, it pulls its neighbors out of their equilibrium locations too, and so on down the line.

The result is that a transverse wave disturbance propagates through the material.

The animation below shows a one-dimensional transverse wave propagating from left to right.

In gases, atoms have no "equilibrium location"; they can move freely past each other. Therefore, if one atom is moved by a disturbance, there's no restoring force to pull it back to its equilibrium location. Transverse waves cannot propagate in gases.

An exception to this is the case of magnetized plasma, which technically is also a gas. In that case a transverse wave can propagate along the magnetic field. It is called an Alfvén wave.