Tinkerbell is a symbol of Jeannette because of the terrible things she endures (as a doll) that are very similar to the things Jeannette must endure as a little girl who's nearly as helpless as a doll.
When the author first mentions her doll, Tinkerbell, she's a symbol for Jeannette in a very literal way. Jeannette has just gotten out of the hospital after having burned herself cooking hot dogs (all by herself, as a three-year-old). Jeannette decides to put her doll through the same thing. She lights a match and holds it to Tinkerbell's face.
Jeannette winds up melting her doll's face, but that doesn't stop her from loving Tinkerbell. She writes, "Even though her face was melted, she was still my favorite toy."
Tinkerbell has been permanently affected by the fire, but Jeannette tries her best to ignore it and still love her. This is symbolic of the way her parents deal with Jeanette's own burns—they try to pretend it wasn't that big a deal.
In the next chapter, as Jeannette's family is "doing the skedaddle," Jeanette realizes she's left Tinkerbell behind. Instead of going back for the doll, her parents drive on.
"Tinkerbell can make it on her own," Dad said. "She's like my brave little girl. You are brave and ready for an adventure, right?"
Tinkerbell becomes a symbol for Jeannette again, because they're both being told they're brave, not because they've displayed any real courage, but because that's what her parents need to believe to be able to convince Jeanette that she's okay. Later, Jeannette herself is temporarily left behind during another of her family's "skedaddles."
At the end of the book, Jeannette actually proves that she is "brave and ready for an adventure" like Tinkerbell was by leaving her parents behind and going off to New York City by herself.
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