If you plot the regression line of y against x, and also x against y, you would get two different lines.

I found it in the middle of teaching a stats class, and feel embarrassed.

I guess normalising is one way to remove the bias.

You are absolutely correct that the difference between y against x and x against y fitting perfectly demonstrates why the bias exists, but the correct way to remove the bias is not normalization, but to use a coordinate-independent regression technique.

See the other comments by many other commenters for details.