Why can't transverse waves propagate through gas or liquid?
What makes you think that transverse waves cannot travel through liquids and gases ?
Yes they can.
No problem with that.
Let me cite a few examples,
Consider, the motion of ripples in a pond. They are transverse in nature (the vibrations are perpendicular to the direction of propagation). And they do travel through a liquid - water.
Also, light is an Electromagnetic wave with well established transverse character - the electric and magnetic fields oscillate mutually perpendicular to the direction of propagation.
However, light travels through most gases and liquids.