[structure that affords passage over a ravine or river] Middle English brigge, from Old English brycge, from Proto-Germanic *brugjo (source also of Old Saxon bruggia, Old Norse bryggja, Old Frisian brigge, Dutch brug, Old High German brucca, German Brücke), from PIE root *bhru "log, beam," hence "wooden causeway" (source also of Gaulish briva "bridge," Old Church Slavonic bruvuno "beam," Serbian brv "footbridge").
The original notion is of a beam or log laid down for crossing. Compare Old Church Slavonic mostu, Serbo-Croatian most "bridge," probably originally "beam" and a loanword from Germanic, related to English mast (n.1). For the vowel evolution, see bury.
The meaning "bony upper part of the nose" is from early 15c.; of stringed instruments from late 14c. The bridge of a ship (by 1843) originally was a "narrow raised platform athwart the ship whence the Captain issues his orders" [Sir Geoffrey Callender, "Sea Passages"].
Bridge in steam-vessels is the connection between the paddle-boxes, from which the officer in charge directs the motion of the vessel. [Smyth, "The Sailor's Word-book," 1867]