Does Edward ever bite Bella? I'm confused about that because she wanted him to so badly in Twilight.
In Twilight, the first book, Edward does not bite Bella except for when he sucks the venom out of her from James' bite. I don't think that's exactly what you mean though. However, if you continue to read the books, a great conflict that occurs through many of them is that Bella wants Edward to bite her, and in doing so turn her into a vampire. Edward is in opposition to this, and argues with Bella repeatedly about turning her into a vampire. Edward does not wish to turn her, whereas Bella wishes to become a vampire. Edward even offers her a deal, that he will be the one to turn her as she wishes, only if she will marry him. Whether Bella takes that deal, and whether she actually becomes a vampire, I'll leave for you to find out as you read on.
I think what you are asking is basically, is it Edward's bite that inevitably turns Bella into a vampire? The answer to that question is no. Edward actually delivers his life-changing venom to Bella, as she lay dying in childbirth, with a syringe of his venom delivered directly to her heart. After that he delivers "bites" at various other pulse points to help his venom spread and speed up the change, hopefully saving her life. I hope this clears up your confusion.
He doesn't really bite her until he is sucking the blood out of Bella so James's venom isn't in her body. Edward could not stop sucking her blood because it was so good to him and he had to stop otherwise he would have killed her.
In the first book of Twilight Edward never bites Bella but sucked the poisonous venom from where James, the tracker who planed to kill Bella, so she wouldn't have to turn into a vampire. I think you might be confused a little with the scene that happened since Edward was having a little difficult time stopping the process of feeding from Bella. He was only saving her. :D
In the book, twilight, edward doesn't bite bella to turn her into a vampire, only to suck the venom back out of her so that she wouldn't become a vampire.
But in the book, breaking dawn, Edward does bite bella after him and jacob have cut renesmee from her. He does this to save her life because she is dying from the pregnancy and the birth of renesmee. But after two days she is alright and a healthy vampire, with aceeding self control and the ability to shield herself and others from mental attacks from other vampires. xx
Edward does in a way bite Bella in Twilight. At the end she is poisoned with venom and Edward bites Bella to suck it out so she won't turn into a vampire.