Why don't ions form crystal lattices in water and other polar solvents?
They actually form: ions can be dissolved up till a certain concentration, and beyond that, they form ionic lattices just as you predict.

The easiest way to describe the scenario as an equilibrium between ions-in-water vs ions-in-lattice: 


If the ion-water interaction is very strong, the ions can be very stable in water, even more than in an ionic lattice, so ions will be in the solution. 
If the ion-water interaction is weak compared to the solid, then the solid phase is much more stable, and only a very low concentration of ions can be in the solution.  


Looking at only interaction energies is a very simplistic view as it does not consider entropic factors, yet can describe at least part of the story. Entropy is generally a huge driving force for any kind of dissolution and allows some ions to be in solution even if the solid state energetically more stable.