Why don't phospholipid bilayers dissolve?
We should first understand what happens when a substance dissolves. During dissolution water interacts with the solute molecule; if the strength of interaction between the molecule and water is higher than the strength of interaction among the solute molecules then the solute dissolves. 
(Also have a look at this post).

Phospholipid is an amphipathic molecule — it has both hydrophobic and hydrophilic parts (this you understand very well). The hydrophobic part is a long chain of hydrocarbon (hence tail) whereas the hydrophilic part is a small but highly polar region (hence head). You should note that these parts are not disjoint but are covalently bonded. 



Even though the hydrophilic part interacts with water (which helps the cell survive in an aqueous environment), the strength of covalent bond (between hydrophilic and hydrophobic regions) is very strong and cannot be broken by physical interaction with water. 
The lipid bilayer (even a monolayer) stays together because the water cannot interact with hydrophobic parts but they interact with each other via a pseudo-interaction called hydrophobic interaction. 

I call it a pseudo-interaction (not a standard usage) because there is not actual molecular interaction that stabilizes the system. Hydrophobic interaction happens via increase in entropy of the system. Read the wikipedia page for details.