If electron drift is slow how can magnetism be a product of length contraction in special relativity?

Question

I've been told both are true, but it confuses me on what is actually happening and what is carrying charge in the wire itself. Length contraction would be almost nothing at the speeds at which electron drift happens. For it to be true I'd think then charge must be carried by something at a very fast speed. There are a lot of electrons moving, the length contraction creates more charge in the same space to the inertial reference frame, and you feel a charge based push perpendicular to the motion of the electrons, is how it is explained to me. I know I'm not understanding something or one explanation was wrong but after looking and not finding any good answers I figured I'd ask here. Thank you for your help in advance.

Answer 1

The length contraction is small but the charge of the electrons in a length of copper wire is huge. So the Coulomb force between parallel wires becomes measurable in a frame moving together with the electron drift velocity.