I've read plenty of arguments stating that Bella Swan is anything but a normal teenager. But, if we skip the whole supernatural love triangle part of the series (Which, granted, is most of it) the question remains: Is Bella Swan a normal teenager?

Lets take a look at the average teenager. More often than not, teenagers are stubborn, make bad choices, and consider themselves the center of the universe. I'm not intending to insult anyone here, this is what I have observed. We don't do it all the time though, other times we can be selfless, make great choices, and happy. We also sometimes go through a "dark" state, such as a state of depression, a Gothic or "emo" stage, et cetera, that can last a very long time or a very short time.
So lets see how Bella fits into these behaviors:

Bad Choices: Skipping college was not a good choice YES

Stubborn: Goes to Forks even though her mother insists that she doesn't have to. YES

Considers herself the center of the universe:I'm sure its there, but it doesn't really show up in the book, so NO

Selfless: Most of the time a selfless act by a teenager is something like watching a younger sibling and skipping a movie with friends, but you could say Bella goes above and beyond in this aspect in New Moon, so YES

Make great choices: I can't see many good choices that Bella makes, most are bad choices disguised as good, so while she may make a few small ones, NO.

Happy: She seems pretty happy when shes with Edward. YES.

Dark State: She was chronically depressed in New Moon after Edward left, so YES.

Bella's average teenager score is 5/7.

Now, we'll take a look at Bella's personality, and go deeper into those common acts by teenagers.
Bella describes herself as a bookworm, with plain brown hair, brown eyes, and looks that are not spectacular. She happily cooks and cleans for her Dad. That takes off a few points, any good Dad would make meals for her, even if he is depressed about his divorce. (How many years after the fact? Shouldn't he have moved on? Maybe Renee was a vampire) As I read the book, found myself picturing her with wire-rimmed glasses and a few blemishes. But at Forks High-School, not one, not two, but three boys display obvious crushes on her. (Is that normal? Wouldn't teenage boys be a little less obvious? Whatever. Thats for another day) SO, as I read I decided she just had some self-esteem issues. Perfectly fine. Events go on. Bella makes a few friends. They go on a trip to La Push. Until She finds out Edward is a vampire, she acts just as slightly nerdy teenager should.

So, if you ignore most of the series, Bella is normal.
But that's ignoring most of the series.

She attracts the attention of Edward Cullen, vampire. Then the attention of Jacob Black, soon to be werewolf. But this article was about her life before the whole vampire thing, so while Bella may not be normal, lots of teenagers aren't normal, almost to the point were it is normal not to be normal.