Bridge Four was originally a bridge crew in the army of Highprince Sadeas. It was regarded as one of the worst bridge crews of which to be a member, due to the remarkably high mortality rate amongst its members compared to that of other bridge crews.
Originally a bridge crew known for its high mortality rate, it was often picked to receive the worst of the new recruits by Gaz and Lamaril. When Kaladin arrived at the Shattered Plains, he was assigned to Bridge Four, and not too long afterward, Kaladin was appointed Bridge Leader. Later, during a particularly bad bridge run, several bridgemen were injured; members of Bridge Four quickly gathered and treated the injured bridgemen. Upon hearing the news of Bridge Four treating its injured members, Sadeas declared that if they could not run, they would not be fed or paid.
Eventually, Bridge Four developed a sense of camaraderie that was not seen within the other bridge crews, and they took to having nightly fires, with soup for all the bridgemen in the crew. Additionally, the members trained with the bridge in their spare time to increase their strength and endurance on bridge runs. After a time, the casualty rate of Bridge Four became far lower than that of the rest of the bridge crews.
To stem the resentment toward Bridge Four from the other crews, Gaz assigned Bridge Four chasm duty as their work detail. Chasm duty involved climbing down into a chasm and collecting armor and weapons from the corpses of soldiers that had died during an assault, and had been washed down by a highstorm. It is down in the chasm that Kaladin performs a kata.
While training with his crew and their bridge, Kaladin decides to try carrying the bridge on its side to use it as a shield to reduce the casualties during a bridge run. While awkward, the crew is able to do this successfully. During a particularly bad bridge run at the Tower, the crew decides to put this maneuver into action. While they didn't lose a single man, the Parshendi fired upon the other bridge crews, bringing them down faster and the assault was a failure due to the innovation of Bridge Four.
Afterward, while on chasm duty, Bridge Four discovered some Parshendi corpses, and formulated a plan to use the found Parshendi carapace as armor and act as a decoy to draw attention away from their bridge crew. In practice, it worked better than expected and there were few casualties on any of the bridge crews that day. In short order, Bridge Four had several members assigned to this dangerous duty. Later, contemplating running, Kaladin then formulated a plan of escape.
During a joint plateau assault with Dalinar, Sadeas betrayed Dalinar and left him stranded on the Tower against a huge Parshendi force. While retreating, Kaladin encouraged the members of Bridge Four to turn back and help Dalinar and his army. Putting their training in the chasms to use for the first time, Bridge Four fought off the Parshendi while Dalinar made his way to the edge of the plateau. After a tough fight against the Parshendi, they managed to escape.
Upon returning to camp, Dalinar confronted Sadeas and demanded to take Bridge Four from him. Sadeas refused. Keeping his word, Dalinar traded his Shardblade for every bridgeman in Sadeas's camp: two-thousand, six-hundred bridgemen, including Bridge Four. They then became part of Dalinar's Army and were no longer slaves.
Following these incidents (since the Cobalt Guard had nearly been wiped out, and the remaining men of the Guard were needed to augment the King's Guard), Dalinar made an offer to Kaladin and members of Bridge Four to become the new Honor Guard. While they accepted the honor of protecting Dalinar and his family, they chose not to be known as the Cobalt Guard and instead to retain the name Bridge Four, thus likely ending the battalion's history.
At the beginning of this second novel in The Stormlight Archive, Kaladin remains the leader of Bridge Four, now charged with protecting King Elhokar and, respectively, the Kholin family as the newly referred to Honor Guard (by outside sources), but the members of Bridge Four still refers to themselves as Bridge Four.
Q. As Dalinar gave his Shardblade to Highprince Sadeas, was his motivation only to free the Brigdemen as "Thank you" for the help? Or did he already plan to build a team from soldiers who are loyal only to him?
A. He certainly saw the side benefits. However, his primary motive was to make a statement. Not just as a thank you, but as a way of proclaiming to all of the Alethi "What we have been doing is wrong. This wealth is not worth the lives of men."