I read that the English army attempted to cross a bridge that only fit two horsemen side by side and that Wallace and the Scots were able to mow them down. Is this actually true? Wouldn't the English has seen how vulnerable they were and couldn't they just turn around once they realized it was a trap?
That's largely what happened. The Scots, led by Wallace and Murray, waited until a decent portion of the English army had crossed and attacked while the rest were still crossing. This meant that the soldiers who had already crossed couldn't retreat and weren't sufficiently large to drive the Scots off. They also had managed to get enough English soldiers on their side that killing them would be a significant loss to the English. The English couldn't simply retreat because it's very hard to get an army to turn around once it's committed that far especially as it would be hard for the soldiers at the rear to know what was happening at the front and if they don't start pulling back everyone in front of them has nowhere to go. It's also worth bearing in mind that Sterling Bridge was basically the only way for an army to march into Northern Scotland so they didn't have an option of going around the Scot's flank.
Blame for the defeat at Sterling Bridge got thrown around a bit at the time it happened and has been thoroughly argued in modern scholarship. Hugh Cressingham, Edward's Treasurer in Scotland, often gets the bulk of the blame. It's claimed that he didn't want to pay for the army to sit in the field and so pushed for an attack against John de Warrenne's desire for caution. Others claim that John de Warrenne was to blame, citing his earlier victory at Dunbar and his general contempt for the Scots they argue that he underestimated Wallace and Murray and simply thought he could take them. In my experience more of the blame gets placed at Cressingham's feet possibly because Warrenne was very well respected and known for competent military strategy while Cressingham appears to have been generally disliked and is best known for being the guy that William Wallace had flayed after Sterling Bridge.
Marc Morris' A Great and Terrible King has a good account of the battle and includes a decent amount of information on John de Warrenne.
G.W.S. Barrow Robert Bruce covers the whole Scottish Wars of Independence from a very Scottish perspective.
Pretty much anything by Michael Prestwich relating to warfare or Edward I covers Sterling Bridge and is of great quality. I quite like Armies and Warfare in the Middle Ages: the English Experience.