This question has an odd component to it. It seems to indicate that Turtle isn't or wasn't the true winner of the Westing Game. By saying that you want to prove she is the true winner, it sends the message that she somehow wasn't. Turtle Wexler absolutely won the Westing Game. She figured out who the fourth was. She figured out that Westing was also Eastman, Sandy, and Barney, and that's why she ended up taking over Westing's company. I don't really think there is an argument to be made that Turtle isn't the winner of the mystery game.
Perhaps a better argument would be to state that Turtle won far more than the game itself and the money that came with the victory, and your decision to focus on Turtle's relationships is a good thought. Turtle matures a great deal over the course of the book. When readers are first introduced to Turtle, it is clear that she has a chip on her shoulder. She is looking out for herself and nobody else. Already in chapter 3, readers are clearly shown how Mrs. Wexler treats Turtle like an afterthought, and by chapter 4, we are shown a Turtle determined to do whatever it takes to get what she wants for herself.
At two dollars a minute, twenty-five minutes would pay for a subscription to The Wall Street Journal. She could stay all night. She was prepared.
As the book and the game progresses, readers see how Turtle begins to soften. We see her take the blame for the bombings. We see how she begins to be more patient with Flora and other people. As for Sam Westing, most of his influence on Turtle comes in the book's final chapters. He trains Turtle to take over the business, but he also trains her to be someone who is devoted to watching out for the well-being of other people. Sam Westing knew what it meant to pay something forward, and the book ends by hinting to readers that Turtle is going to do the same thing with Alice.
"Hi there, Alice," T. R. Wexler said. "Ready for a game of chess?"
Turtle becomes the "true" winner because she won the game while becoming a better person.