Bella would likely be very frightened during the car incident. Once Edward saves her, she'd be extremely relieved and probably feel even more attached to him. Edward, on the other hand, would be initially worried and focused on getting Bella to safety. After saving her, he might feel a sense of relief as well, but also a bit of that over - protective feeling that he often has for Bella.
Well, in the 'Edward Saves Bella from Car Fanfiction', it could start with Bella in a car that has some sort of mechanical failure or is in an accident caused by another vehicle. Edward, being attuned to Bella's presence, senses her danger. He rushes to the scene. If it's a collision situation, he might shield Bella from the impact with his body. Or if the car is on fire, he gets her out safely without getting burned himself due to his vampire nature.
She might be in shock at first. After all, she was in a life - threatening situation. But then she could feel extremely grateful towards Edward and their relationship might grow stronger.
In many such fanfictions, Edward might use his superhuman speed to reach Bella just in time. He could have sensed her danger through his vampire instincts and dashed to the cliff edge, grabbing her before she falls.
In many Twilight fanfictions, the scenario of Edward saving Bella from Jacob can be quite intense. It often plays on the complex love triangle dynamic. Edward might save Bella from Jacob in a physical altercation, perhaps when Jacob's wolf instincts get out of control. This could be a way to reaffirm Edward's role as Bella's protector in the eyes of the fans who are rooting for their relationship.
Edward could use his mind - reading ability. He reads Charlie's intentions and plans his moves accordingly. If Charlie is about to make a decision that Edward believes will be bad for Bella, he can intervene at the right time. He might also be able to manipulate the situation subtly so that Charlie changes his mind without realizing it.
One possible reason could be that Edward believes Charlie is a threat to Bella in some way that Charlie doesn't realize himself. Maybe Charlie unknowingly gets involved with some dangerous elements, and Edward, with his vampire senses, detects it.
One key element is the shift in power dynamics. Usually in the original Twilight story, Edward has a lot of control over Bella. In these fanfictions, Jasper disrupts that by saving Bella, which changes the whole dynamic. For example, Bella might become more independent.
In some plotlines, it could be that Bella has made a deal with Peter in order to escape Edward's over - possessive nature. Peter might have his own reasons for wanting to get involved, like a long - standing feud with Edward's family. So, he uses his wits and resources to whisk Bella away to a place where Edward can't find her. This could lead to a whole new adventure for Bella as she adjusts to life away from Edward and with Peter's protection.