Social media has affected people's information sources. Since most of the news on social media is not verified by a central authority, it may contain fake news for various reasons such as advertising and propaganda. Considering an average of 500 million tweets were posted daily on Twitter alone in the year of 2020, it is possible to control each share only with smart systems. In this study, we use Natural Language Processing methods to detect fake news for Turkish-language posts on certain topics on Twitter. Furthermore, we examine the follow/follower relations of the users who shared fake-real news on the same subjects through social network analysis methods and visualization tools. Various supervised and unsupervised learning algorithms have been tested with different parameters. The most successful F1 score of fake news detection was obtained with the support vector machines algorithm with 0.9. People who share fake/true news can help in the separation of subgroups in the social network created by people and their followers. The results show that fake news propagation networks may show different characteristics in their own subject based on the follow/follower network.