The COVID-19 virus has become a fearful epidemic for people all over the world. In Turkey, long quarantine periods and curfews have increased both physical and psychological problems. Due to the rapid spread and substantial impact of the COVID-19 virus, different psychological effects were observed among different segments of society, such as among young people, elderly people, and active workers. Because of fear caused by the COVID-19 virus, it is thought that depression, stress, and anxiety levels have increased. It is estimated that there are more psychological issues for people with poor health and others whose friends or family became ill or have died because of COVID-19. To explore and test the situation mentioned above, we conducted a cross-sectional study in Turkey with 3287 participants above 16 years old. We measured COVID-19 fear, along with anxiety, stress, and depression levels (DASS21) and demographics. Firstly, we tested whether COVID-19 fear predicts stress, anxiety, and depression. Secondly, we investigated if the effect of COVID-19 fear is stronger for those who have underlying illness and for those whose friends or family became ill or have died because of COVID-19. The results showed that women and 16-25 years old youths have higher COVID-19-related fear, anxiety, depression, and stress. Furthermore, we found a significant relationship between COVID-19 fear and stress, anxiety, and depression, as well as significant moderation effects of having an underlying illness and having friends or family who were infected or have died. These results show the importance of implementing specific implementations, particularly for vulnerable groups, to minimize the psychological problems that may arise with the pandemic.