The conflict in the novel is primarily racial. Even though the text begins much later, it has it's roots with the Logan's acquisition of part of Mr. Granger's land which he had to sell during reconstruction to cover the taxes for the rest of his land.
Historically speaking, land symbolized political emancipation and agency in America. At the founding stages of the country, only land-owning, white males were allowed to vote.
The Logan family's land thus gives them roots, history and symbolic rights. The conflict arises because whites attempt repeatedly to undermine those rights, but they do so in subtle ways and quite often outside the confines of the law. The incident when Cassie bumps into Lillian Jean is a good example. In the 1930, even the the Jim Crow Laws were quite intact and segregation well enforced, there was certainly no written protocol to follow when a black girl accidentally bumps into a white girl on the sidewalk. Mr. Simms gets involved in what is essentially a scuffle between two children. He literally puts Cassie in her place by pushing her into the road, something he would never do to a white child. The racial conflict is thus overarching and pervasive in the narrative, but this incident also illustrates that it grows out bounds and breeds a hatred that knows no boundaries. Other examples are the burning of Mr. Berry and the threatened lynching of T.J.
The novel's conflict is a family going through rough times in Mississippi is facing racial discrimination and the children don't understand why they are being treated unfairly.