Since this is a book for young adults, intended to hold their interest, each chapter is episodic. That is to say, within each chapter, which represents the children's visits to their grandma's house, there is some type of lesson learned, a reason for the events to unfold. As the book progresses, Joey and Mary Alice learn to respect, love, and admire their eccentric grandmother. I think you could make a case for the turning point occurring when the children learn about prejudice and the adult world. But remember, as a book that travels over the time span of the years in a child's growth, there will be more than one turning point!
Turning Point- The point at which a very significant change occurs; a decisive moment.
The turning point in A Long Way from Chicago is when the characters of Mary Alice and Joey complete their change from being wary and uncertain of Grandma Dowdel's actions to emulating them without provocation.
In the chapter The Phantom Brakeman, Mary Alice acts like Grandma when she intervenes and helps Vandalia Eubanks by hiding her upstairs. Later in the chapter, it is inferred that Joey completes an eloborate ruse, posing as the Phantom Brakeman, to help Vandalia and Junior Stubbs get on the train escaping their families. The children's decision to help others in need because it is "the right thing to do" marks a significant change for their characters. Up to this point in the story, Grandma Dowdel did all of the "intervening," but now the children have adopted Grandma's sense of right and wrong and have acted definitively.