This isn't a new narrative really. It's a classic fairy tale formula - a woman, either an average pretty girl or an actual princess - meets a prince who takes her to his castle, dresses her up in fine clothing and either makes her royalty or makes her qualified to rule through marriage. Either way, she is improved merely by being with him.
Now we do see movies where the opposite is true, a woman strives to make her boyfriend or husband shape up, takes him shopping and buys him nicer clothes, and motivates him to get a job. However, the woman who does this is typically framed as a bitch, or a ball-buster, the guy complaining that her efforts to shape him are emasculating. Of course, the woman usually only intervenes in his life if he's in really bad shape and needs her help - unemployed, dirty irresponsible, plays video games all day long and dresses like homeless man. Rarely does the female character intervene when he's doing all right.
The girl improvement narrative, which I am hereby coining on my own, is a little different. Rather than a woman improving her male partner with her femininity, her partner improves her femininity. He takes her in and improves how she dresses, how she talks, how she carries herself, etc., even if she's doing fine on her own This is often seen in Disney movies, but Twilight takes it to a new degree, trading the sparkling tiara for glittering skin.
Bella draws a lot of criticism for being bland, clumsy, lacking in ambition and any real ability to make up her mind. However, had she not found Edward she probably would have grown out of her awkwardness and matured, personally I give the girl a break for being human and only a teenager. But that's just it, she's only human. Edward swoops in and saves her from her human flaws, and in fact, her very humanity itself. Before they marry he gently works to "improve" her, practically making her apply to various schools, helping her with her homework, in the book he put shiny things in her hair for prom. Though I shouldn't be too harsh on him, at least he doesn't criticize her character flaws or teach her how to walk, sit, and speak like in the Princess Diaries.
The Cullens are like royalty, their mansion a castle. They are wealthy, well known in school, accomplished and well spoken. Bella is afforded every luxury when she is with them - a glamorous birthday party (and later a lavish wedding), beautiful dresses to wear, a luxurious bed to sleep on when she has to spend the night. The couple spend their honeymoon on a gorgeous private island. This is all in stark contrast to her human life, where she lives in a small house and takes care of her father - if this is starting to remind you of Beauty and the Beast, you're not alone. It really is like a modern rags-to-riches fairy tale, where her life is drastically improved just by being his girlfriend. This is not always addressed when critics point out how incomplete she is without him.
Finally, after they marry and she dies in childbirth, almost ruined by the result of their partnership and intimacy, he turns her into a vampire, giving her new life. This is also rarely addressed in the common critiques about the abusive relationship she has with Edward. After enduring his controlling nature for over a year, she finally commits to him forever, gives him her virginity and as a result wakes up covered in bruises with a monster growing inside her. The birth, which to her credit she does choose, kills her. This may remind the reader of the way abusive partners destroy the souls and selfhoods of their victims so they can build them anew, making their victims into what they want them to be. In Breaking Dawn, the Bella that Edward creates by turning her into a vampire is beautiful, sparkling, coordinated and strong - just like the rest of the Cullens. Not to mention that in becoming a vampire, she is forced to cut off all contact with her father and her human friends, and she isn't supposed to be friends with Jacob but at least he is in on the secret so eventually it's okay.
But this isn't about the escalating abusive nature of the relationship. This piece is about how by being his, Bella is transformed from a normal girl to a dazzling princess- I mean vampire. This age old narrative has drawn criticism for the messages it sends to girls, but I want to focus on the message it sends to guys and the behavior it normalizes. These stories subtly tell guys that it is their job to improve the women that they're with, rescuing them from their mere mediocrity. I've known young men who have internalized this, one who did feel it was his job to have a paternalistic relationship with all the women in their lives, giving them advice and making them "better" whether they wanted his help or not. When I rejected his unsolicited guidance he vilified me for being ungrateful, saying he wouldn't help me at all ever again. To him it was absurd that I wouldn't want to improve (I did, I just wanted the improvements to be of my own doing), that by being happy with who I was I was merely settling, and that a mature woman would be happy to accept his improvements.
This used to be the rationale for allowing men to hit their wives. In fact, it is still the rationale behind Christian Domestic Discipline marriages. The idea is that women require guidance from their husbands, and that by correcting bad behavior through spanking, he improves her very character.
The "girl improvement" narrative in various fairy tale, rags to riches movies only normalizes the condescending, paternalistic, borderline controlling way too many guys treat their wives and girlfriends, acting as though it is their duty to make them better women. Twilight is by no means the only example of this narrative that women need to be improved by their male partners, it's just a glaring example.