Infant and childhood growth are dynamic processes characterized by drastic changes in fat mass and body mass index (BMI) at distinct developmental stages. To elucidate how genetic variation influences these processes, we performed the first genome-wide association study (GWAS) of BMI measurements at 12 time points from birth to eight years of age (9,286 children, 74,105 measurements) in the Norwegian Mother and Child Cohort Study (MoBa) with replication in 5,235 children (41,502 measurements). We identified five loci associated with BMI at distinct developmental stages with different patterns of association. Notably, we identified a novel transient effect in the leptin receptor (LEPR) locus, with no effect at birth, increasing effect on BMI in infancy, peaking at 6-12 months (rs2767486, P6m = 2.0 × 10-21, β6m = 0.16) and little effect after age five. A similar transient effect was found near the leptin gene (LEP), peaking at 1.5 years of age (rs10487505, P1.5y = 1.3 × 10-8, β1.5y = 0.079). Both signals are protein quantitative trait loci (pQTLs) for soluble LEPR and LEP in plasma in adults and independent from signals associated with other adult traits mapped to the respective genes, suggesting novel key roles of common variation in the leptin signaling pathway for healthy infant growth. Hence, our longitudinal analysis uncovers a complex and dynamic influence of common variation on BMI during infant and early childhood growth, dominated by the LEP-LEPR axis in infancy.