THE CURRENT GETS TO A BULB NEAREST TO THE POSITIVE TERMINAL FIRST
WRONG!!
We DO say that current flows from the positive terminal to the negative terminal but this is just the direction.
Wires and the filaments of light bulbs are full of electrons which are free to move when given a force, so when the switch is closed and the circuit is complete EVERY electron has a force on it at the same moment...so the current starts flowing EVERYWHERE in the circuit at the same time
The comparison is with a hose of water. Imagine a horizontal hose FULL of water connected to a tap (turned off) at one end and open at the other. As soon as the tap is turned on (like the switch),
there is a pressure difference across the water from the tap to the open end (like a voltage difference across a wire from the positive to negative). Remember the hose is FULL of water (like the wire is full of electrons) so the water starts to move at all points along the hose at the same time.
To confuse the issue the electrons do not go the same direction as the current, but I shall not explain that here.