Did fascism and communism grow after World War I?

Both did.

Fascism grew because it is basically extreme nationalists. This would appeal to those who came out strong in World War 1.

Communism also did, for those who lost the war. You see, the citizens of a defeated country feel as though they have been used and not recognized or appreciated. However, if they turn to a communist society, then everyone will be one and will be equally accepted and recognized.

Hope this helps, as I just finished studying World War 1 myself.