That's true, but can shear waves be transmitted to the earth's solid core through the liquid iron surrounding it?

Yes. And to add a missing piece, seismic waves radiate out from the point of vibration. Some of these waves pass through the core of the earth and are received on measurement equipment on the other side. Some of these waves pass through the mantle or the crust. If we know the epicenter of the quake, then we can estimate how long these waves should take to reach our different measuring stations assuming different densities.

Given that we can know how long they actually do take, we can turn around and estimate the mean density along the straight-line path they took. When we determine that the mean density is significantly higher for paths through the center of the earth, that tells us that the material there is much denser, and so we suppose that it could be solid.