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.
Maybe just think of it as the electrons hitching a ride through the electrolyte on the relatively giant and mobile atoms that make up ions in the solution. If you really wanted to force those electrons to travel on their own through the electrolyte you'd have to crank up the voltage big time. Thousands of volts would be needed to provide the electrons with enough energy to rip through the electrolyte (i.e. dielectric breakdown). 

Another way to get electricity to flow through an electrolyte, although still not electrons flowing, is to impose an AC current across a cell. At high enough frequency (usually in kHz range) current can short through the capacitance at each electrode. This is typically how conductivity measurements are made on liquids.