Question: I was learning about galvanic cells and I had a problem understanding why electrons do not travel through the electrolyte solutions themselves, instead preferring to travel through metals. Can electrons travel through an electrolytic solution?

Imagine a galvanic cell, without the wiring between the two electrodes and instead we only have the salt bridge. Won't the electrons (although it is very unfavorable to happen) travel through the salt bridge and the solutions? In other terms will the transformation of the electrodes occur?

Answer: 
Not in water. Free electron in water is really unfavorable, so no significant concentration of them can be generated chemically, and it almost immediately reduces water itself to hydrogen (but I heard rumors about generation of solvated electrons in water in very special experiment with short half-life)

In liquid $\ce{NH\_3}$, however, solvated electrons can occur, so self-discharge of galvanic cells with $\ce{NH\_3}$-based electrolyte may occur through travel of electrons via electrolyte.
Electron's can travel through the electrolyte solution, however electrons take the pathway with the lowest resistance, the solution has a relatively higher resistance compared to the outer circuit. Hence the electron takes the path of the outer circuit.