Turbosuperchargers have come a long way in terms of reliability, economy, and driveability.
Besides the technical points with have been amply posted, they are small, easy to adapt to most crowded engine compartments (by the manufacturers as well as rodders), drive is easier than the mechanicals, and relatively cheap.
I have read about units which were tested in the old days which worked on the exhaust side. It wouldn't surprise me to see some manufacturer resurrect exhaust side units as 'latest technology', although I think the emissions implications would be hard to cover up.