There are subtle differences.
"Go through" emphasizes the experience of the difficult situation. "I am going through a divorce".
"Get through" emphasizes the completion of the difficult situation. "It will be difficult, but I will get through this divorce".
"Get by" focuses on the economic aspect and suggests just barely surviving the experience. "How will you manage without the extra income? I will get by somehow."
"Get over" is usually used in the expression "Get over it." which is what you say to someone when you are tired of their (emotional) complaints about a difficult situation. "It is hard for me to accept that I will never see her again. Get over it!" It is generally not a nice thing to say, but you could say it to a friend to get them to move on.
There are probably other subtleties, but these are the ones that first come to mind.