Age at first sex (AFS) is a key indicator for monitoring sexual behaviour risk for HIV and other sexually transmitted infections. Reporting of AFS data, however, suffers social-desirability and recall biases which obscure AFS trends and inferences from the data. We illustrated AFS reporting biases using data from nationally-representative Demographic and Health Surveys conducted between 1992 and 2019 in Ethiopia, Guinea, Senegal, and Zambia. Based on this, we proposed a time-to-event, interval censored model for the distribution of AFS that uses overlapping reports by the same birth cohort in successive surveys to adjust for reporting biases. The three-parameter log-skew-logistic distribution described the asymmetric and nonmonotonic hazard exhibited by empirical AFS data. In cross-validation analysis, incorporating a term for reporting bias as a function of age at report improved model predictions for the trend in AFS over birth cohorts. In the four example applications, the quartiles of the AFS distribution were 16 to 23 years for Ethiopian and Senegalese women and 15 to 20 years for Guinean and Zambian men. Median AFS increased by around one to 1.5 years between the 1960 and 1989 birth cohorts for all four datasets. During adolescent and young adult ages, men tended to report an earlier AFS while women tended to report an older AFS than when asked in their late twenties. Above age 30, both male and female respondents tended to report older AFS compared to when surveyed in their late twenties. Simulations validated that the model recovered the trend in AFS in the presence of reporting biases. When there were biases, at least three surveys were needed to obtain reliable estimate for a 20-year trend. Mis-specified reference age at which AFS reporting is assumed unbiased did not affect the trend estimate but resulted in biased median AFS in the most recent birth cohorts.