A 4:3 DVD has the same number of pixels as a 16:9 DVD. There is a flag on the DVD that tells the TV how much to stretch the pixels horizontally. There are never any black bars on the DVD per se.
Playing a 4:3 DVD on a widescreen (16:9) TV, black bars are placed left and right by the DVD Player.
Playing a widescreen (16:9) DVD on a 4:3 TV, black bars are placed top and bottom by the DVD Player.
It seems as someone rendered your DVD with the black bars imposed on the image on the DVD, which is not the proper way. There isn't anything you can do about that: You can't get back the pixels lost in the process.