How does a solvent affect the solubility of an ionic compound?
Well, the number of solvents that are capable of solvating ionic solutes is fairly small.
Water is an exceptionally powerful solvent, and MOST solutes, ionic or not, have SOME solubility in an aqueous medium to a greater or lesser degree.
Water is a polar, protic solvent that is capable of solvating both positive ions and negative ions. When we dissolve a salt, say common salt in water, we write...
On the other hand, organic solvents, ethers, alkyl halides, even alcohols, have next to no capacity to dissolve ionic solutes...because these solvents are grossly incompetent to solvate ions.