Little is known about the impact of the COVID-19 pandemic on the mental health and psychological well-being of young people. The aim of this study is to investigate the psychological well-being and changes in the mental-health state of young people living in Austria and Turkey. By using an anonymous online survey, we recruited 1240 people aged 15–25 years from these two countries. We used the “Psychological General Well-being” and a self-created questionnaire to capture individual experiences during the COVID-19 pandemic and quarantine period. The native Turks indicated higher “anxiety” (ps < 0.010), lower “vitality” (ps < 0.011), and lower “general health” (ps < 0.011) than native Austrians or Austrian migrants and increased “depression” (p = 0.005) and lower “self-control” (p = 0.022), than Austrian migrants. Moreover, 50.9% of native Turks reported a decrease in their mental health status, compared to 31.1% of native Austrians and 23.7% of Austrian migrants. Participants with financial problems (OR = 1.68) and prior mental health problems (i.e., already in treatment by the time of COVID-19, OR = 5.83) reported a higher probability for a worsening in their mental health status. Our results show that the COVID-19 pandemic impaired the psychological well-being and mental health of young people. Especially people in Turkey were most affected, probably due to the stringent policies to fight COVID-19.