Yes, I think that a woman who was able to overcome what Natalie did and still find joy and be able to her goals is a strong woman. I was surprised by a few things in the ending, but overall was quite pleased with Natalie's later life.
No child should have to develop their character through the types of experiences that Natalie faced. Despite what she faced she demonstrates courage to defend her mother in spite of her fear for her stepfather. She also shows perseverance by choosing to stay close to her mother when future violent outbursts were imminent. In the prologue we see both Natalie's strength of character, and the root of her main character flaw, her bitterness. The author does a great job of encapsulating the conflict of the entire book through this brief prologue.
Being the oldest child in the family and feeling the need to protect them probably forced her to grow up a lot faster and to be stronger than she might have been under other circumstances.
The strength of a character is measured by their choices. A situation alone doesn't define strength but how they react to that situation determines how much strength a character has. In the Prologue of this story already get to see Natalie's strength in her response to her mother's getting beaten. She responds in rebuking her step-father and declaring that he stop. And, in this moment, he listens. The only beating where grief brought out the violence instead of alcohol. Of course, there are many different strengths, so we will still get to see Natalie grow in other ways. And then, there are a few character traits where Natalie suffers a fall of sorts as her situation molds her in a negative manner. Perhaps a stronger character might refuse to let the situation corrupt her, but it's already tough enough just survive an abusive father-figure.
“History doesn’t repeat itself, but it does rhyme” - Mark Twain. Dare we say the same thing about every story that gets told in the world?