Why is inductance (L) proportional with turns-square (N²)?
Think of a single turn inductor (left below) then, imagine that single turn split into two parallel wires that are wound very tightly so that they occupy virtually the same space (right below). 

The two parallel wires, for a given applied voltage, will each take half the current of the single turn inductor and, together they take the same current as the single turn: -

 

Because of this, each individual parallel wire MUST have twice the impedance of the single wire and, together, when wired in parallel, exhibit the same impedance as the single wire. OK so far?

Now, re-arrange those two wires (in your mind's eye) so that they are in series with each other. The impedance changes to four times the impedance: -



This means the inductance has quadrupled for a doubling of turns and it is trivial to extend this example to n turns.