I am reading about electrolysis and I have trouble to understand why an electrolyte must be added in order for electrolysis to occur. I have read this [question](https://chemistry.stackexchange.com/questions/60737/how-do-electrolytes-allow-electrolysis-to-occur) but still can't get it. The definition of electrolyte is a substance that when dissolved produces an electrically conducting solution.

**Electrolytes in electrolysis**

Why we need an electrolyte in order for electrolysis to occur? In electrolysis of water the water is consumed not the electrolyte. So why we need the electrolyte? Couldn't the reaction happen without the electrolyte?

After reading the comments I have modified the question. As I said before I can't understand the purpose of the electrolyte. For example in the electrolysis of water with sulfuric acid why we add it? Water molecules are in contact with both anode and cathode. So why we need to add the electrolyte?

What happens at molecular level when the electrolyte is added? How the ions move in the solution (if they move at all)?