The introduction of long-chain branches can significantly increase the melt strength and processability of the polyesters. Hence, in the present study, a number of long-chain branched copolyesters were synthesized and the effect of branching agent on the properties of copolyesters was examined. Pentaerythritol (PER) and trimellitic anhydride (TMA) were used as branching agents for the synthesis of poly(butylene succinate-co-ethylene terephthalate) (PBSET). Microstructure and composition of the copolyesters were characterized by 1H. NMR and their successful synthesis were corroborated. DSC test proved the semi-crystalline nature of copolymers and corroborated the crystallinity decrement with branching. The crystallinity decreased by 30–47%, when long-chain branches were formed in PBEST. Interestingly, no secondary crystallization was observed using the Avrami model. Furthermore, the Avrami exponent was in the range of 2.5–4.5, signifying a 3D-crystal growth. According to the shear viscosity measurement, the branched copolymers revealed more shear thinning behavior compared to their linear counterparts, and according to the elongational viscosity measurement, the PER branched copolymer displayed a stronger strain hardening response compared to its linear and TMA branched counterparts. Moreover, the shear modulus was raised by two orders of magnitude with branching. Having higher entanglement and less mobility, the long-chain branched copolyesters displayed longer relaxation times compared to their linear counterpart. Despite the outstanding feature of the TMA, its inclusion more than 0.4% (per mol) was not possible due to its declining effect on copolymer extensibility.