One of the conflicts in the novel is the present idea of the American Dream but the impossibility of its realization. Some say that the American Dream is something like a "rags to riches" story or the idea that with hard work, one can achieve success. Although these ideas seem quite possible, it is not a given that one will always achieve the dream. In addition, when one engages in illegal or unethical means to achieve the American Dream, that dream is corrupted and not nearly as idealistic in its purest sense.
In The Great Gatsby, Gatsby engages in illegal means (gambling, bootlegging) in order to achieve some of his wealth. He also inevitably doesn't win Daisy back. Part of this conflict with the elusive American dream is the conflict between the idealistic Midwest (where Gatsby/Gatz, Daisy, and Nick are from) and the more sophisticated but also more corrupt East (the city) where Gatsby ends up trying to manifest his American Dream.
There are a few moments where Nick realized overtly that Gatsby's American Dream is impossible to achieve. After Gatsby professes his love of Daisy in front of Tom, Nick senses that the dream is "dead" as Gatsby continues to plead his case/love:
But with every word she was drawing further and further into herself, so he gave that up, and only the dead dream fought on as the afternoon slipped away, trying to touch what was no longer tangible, struggling unhappily, undespairingly, toward that lost voice across the room.
In the closing moments of the novel, Nick notes that Gatsby's dream was not just one of an impossible future; it was also the naive impossibility of trying to recreate the past. Of Gatsby, Nick says:
He had come a long way to this blue lawn, and his dream must have seemed so close that he could hardly fail to grasp it. He did not know that it was already behind him, somewhere back in that vast obscurity beyond the city, where the dark fields of the republic rolled on under the night.