Background The COVID-19 pandemic represents a global health crisis. The Shanghai municipal government in China implemented strict and comprehensive pandemic control strategies in the first half of 2022 to eliminate a wave of COVID-19 infection. The pandemic and the resulting government responses have led to abrupt changes to families’ daily lives, including the mental health of children and adolescents. Objective The aim of this paper is to examine the impact of COVID-19 exposure and the stringent lockdown measures on the daily life and mental health of children and adolescents and to provide suggestions on maintaining their mental health when similar public health emergencies occur in the future. Methods In this cross-sectional study, an anonymous survey was distributed online in May 1-15, 2022, in Shanghai. Individuals were eligible to participate if they were currently the caregiver of a child or adolescent (aged 4-17 years). Outcomes were psychosocial functioning of children and adolescents, as reported by parents, using the Pediatric Symptom Checklist-17. COVID-19 exposure and life changes were also reported. Multivariate logistic regression was used to analyze risk factors for poor psychosocial functioning. Results In total, 2493 valid questionnaires were analyzed. The rate of positive scores on the global Pediatric Symptom Checklist-17 scale was 16.5% (n=411). Internalizing, attention, and externalizing problem subscale positivity rates were 17.3% (n=431), 10.9% (n=272), and 8.9% (n=221), respectively. Caregivers reported that 64.2% (n=1601) and 20.7% (n=516) of the children’s interactions with friends or peers and parents deteriorated, respectively. Compared with male caregivers, female caregivers were less likely to report psychosocial problems in children and adolescents (adjusted odds ratio [aOR] 0.68; 95% CI 0.53-0.88). Older children and those with lower COVID-19 Exposure and Family Impact Scales scores were less likely to have psychological problems (aOR 1.15; 95% CI 1.10-1.21). Compared with children with screen times <1 hour per day for recreation, those using screens for >3 hours had higher odds of psychological distress (aOR 2.09; 95% CI 1.47-1.97). Children who spent 1-2 hours exercising and had better interactions with friends or peers and parents showed a trend toward lower odds of psychological problems. Children and adolescents with worse sleep compared with preclosure were more likely to have psychological problems. Conclusions The prevalence of psychosocial problems among children and adolescents is relatively high. Being young, having more COVID-19 exposure, and having more screen times (>3 h/day), less exercise time (<30 min), worse sleep, and deteriorated interactions with friends or peers and parents were risk factors for poor psychosocial functioning. It is necessary for governments, communities, schools, and families to take appropriate countermeasures to reduce the negative impact of the stringent control measures on caregivers’ parenting and psychosocial functioning of children and adolescents.