Propagation of waves depends upon the medium through which they propagate. To understand this one needs to understand the phenomenon of interaction of particles in a given medium. read more
Thus, transverse waves are possible in solid elastic medium, but they are not possible in fluids as fluids do not have the necessary shear elastic property. read more
For the propagation of a transverse wave restoring force is perpendicular to the direction of particle vibration.For a taut string this restoring force is provided by tension in the string.For solids this is provided by shear modulus. For liquid shear modulus is zero ,so no transverse wave. read more
To answer your question, first let's see how transverse waves or S-Waves propagate, so in simple terms we can state that, S-Waves or transverse are shear waves, whose particles move perpendicularly to their direction of propagation. Now, let's see why can they propagate through solids. read more