Originally called the Bridge of Stonebows, it was built during the days of power of Arnor. Marcho and Blanco with their Hobbit followers passed that bridge to settle in the Shire. The Kings of Arnor required them to keep the Great Bridge in repair (as well as all other bridges and roads).
Frodo Baggins and his companions did not use the bridge when they left the Shire in 3018. However, on their return trip they came to the bridge on 30 October 3019. To their surprise they found the bridge blocked on both ends with two spiked gates. Despite a sign proclaiming "No admittance between sundown and sunrise" the four hobbits made their way over the bridge, after dispatching Bill Ferny. They stayed the night at the guard-house before moving on to rouse the Shire.
In Fo.A. 15 King Elessar rode to the Brandywine Bridge and greeted his old friends (but did not cross into the Shire for he himself had forbidden men to enter the land in an edict of Fo.A. 6). At this meeting he bestowed the Star of the Dúnedain on Samwise Gamgee and made Sam's daughter Elanor a maid of honour to Queen Arwen.