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: 
You're questioning the intuitive disconnect caused by most galvanic cell drawings which seem to assume the electrolyte solution in the salt bridge does not conduct electricity, so let's investigate.

Imagine a Zn/Cu$^{2+}$ cell with electrodes 5 cm apart in a 3.5% NaCl solution with a tube (1 cm$^2$ cross-section) of solution as the salt bridge for balancing charge.

The electrical resistance (R = $\rho$l/A) of our NaCl 0.05 m x 1 cm$^2$ salt bridge solution is: 

$$\frac{0.2 \ ohms\*m}{} \* \frac{0.05 \ \* m \ (length)}{10^{-4}m^2 \ (cross-section \ area)}= 100 \ ohms$$

Considering the predicted EMF of 1.1 Volts for this cell, the expected current ($I = V/R$) through the salt bridge is: $1.1V/100 \ ohms \ = 0.011 \ amps$

This current may be negligible in a galvanic cell drawing compared to the current through some wire or low-resistance load. However, this would make a terrible battery for most common purposes as a typical AA battery (3000 mAh) would go completely dead in less than 2 weeks if it actually leaked at this rate!

It seems then your intuition is basically right... until you understand what the models leave out. In real alkaline battery designs, the cathode, electrolyte, and anode are sandwiched together very closely with a very large surface area, yielding excellent conductivity through electrolyte (and therefore very low resistance). However, these layers are separated by a membrane which allows ions through but has a very high resistance to electric current.

Sources:
<https://www.thoughtco.com/table-of-electrical-resistivity-conductivity-608499> (seawater resistance)
<https://en.wikipedia.org/wiki/Alkaline_battery> (alkaline battery design)
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.