Can liquids exert/be affected by normal force?

I am looking for a force similar to the normal force, but as far as I can tell, the normal force can only exist between two SOLID objects, not a solid and a liquid.

Not quite; liquids can also apply/receive a normal force. The catch is that additional normal forces must act in the other two directions as well. (For example, a normal force in the x-direction requires normal forces in the y- and z-directions.) The reason is that the ideal liquid can't sustain a shear stress, and all other stress states than equitriaxial stress (the compressive version is called hydrostatic stress, also known as pressure) would result in a shear component. Thus, only equitriaxial loading is possible at equilibrium.

If we consider gravity, then the normal forces on the bulk liquid aren't all the same (because the body force of the weight represents an additional load that breaks the cubic symmetry), but each infinitesimal element of the liquid is still under equitriaxial stress.

With that constraint met, there's no problem in assigning normal forces; one example would be the force that pushes back when you try to compress a liquid in a container.