Does making the nose bridge narrower also make it higher?

You can obtain some elevation with the narrowing of the bridge.Find a board certified plastic surgeon who performs hundreds
of rhinoplasties and rhinoplasty revisions each year. Then look at the
plastic surgeon's website before and after photo galleries to get a
sense of who can deliver the results. Kenneth Hughes, MD Los Angeles, CA

 When medial and lateral osteotomies are performed as a stand-alone procedure, there is a slight elevation of the mid bridge area. If a significant augmentation is needed, then a cartilage graft will be required to the nasal bridge in addition to the osteotomies to narrow the bridge line. For many examples, please see the link below

If you have a low nasal bridge, then you probably need a dorsal onlay graft. Osteotomies narrow the nose and do not really raise the bridge.