Infant and child mortality are often used to monitor the progress of national population health programs. The data for this study was collected from selected urban slums where icddr,b has maintained the Health and Demographic Surveillance System (HDSS). Using the HDSS database, 6,666 married women were selected and interviewed in 2018 to collect data on socioeconomic status, pregnancy history and safe motherhood practices. The study examined levels and trends of infant and under-five mortality for three periods: 1990–1999 (Period 1), 2000–2009 (Period 2), and 2010–2018 (Period 3) and examined socio-demographic differentials of infant and under-five mortality for Period 3. From Period 1 to Period 3, under-five mortality declined by 68.2%, with child mortality (1–4 years) declining more than infant mortality (84% vs. 65%). In the regression models for Period 3, infant and under-five mortality were higher for working than non-working mothers (infant: OR = 1.35*, CI: 0.98, 1.86; under-five: OR = 1.34*, CI: 0.99, 1.82), lower for girls than boys (infant: OR = 0.77*, CI: 0.57, 1.03; under-five: OR = 0.77*, CI: 0.58, 1.03), higher for small-size than normal/big-size babies (infant: OR = 4.11***, CI: 3.00, 5.64; under-five: OR = 3.68***, CI: 2.70, 5.02), higher for babies delivered vaginally than by caesarean section (infant: OR = 1.79**, CI: 1.14, 2.97; under-five: OR = 1.87***, CI: 1.21, 2.88), higher for babies delivered with complications than no complication (infant: OR = 2.16***, CI: 1.48, 3.15; under-five: OR = 2.21***, CI: 1.55, 3.18), and higher for babies born after a short (<24 months) birth interval (infant: OR = 1.71*, CI: 0.96, 3.05; under-five: OR = 1.63*, CI: 0.93, 2.86) than firstborns. While substantial progress has been made in reducing under-five and infant mortality, neonatal mortality have declined less slowly. Targeted population health interventions addressing the socio-demographic drivers of infant mortality, with a focus on the urban poor, will help Bangladesh achieve Sustainable Development Goal 3.