Why don't ions form crystal lattices in water and other polar solvents? Ionic compounds are in the form of crystal lattices, and when they are dissolved in a polar solvent, they are bonded with the corresponding cations/anions of the polar solvent. For example, $Na^+$ bonds with $OH^-$ and $Cl^-$ bonds with $H^+$ when NaOH is dissolved in $H_2O$. 

The reason why $NaCl_{(aq)}$ conducts electricity is because $Na^+$ and $Cl^-$ do not form crystal lattices with $OH^-$  and $H^+$  respectively. But why don't they form crystal lattices?
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.