To solve the problem of chromatographic separation encountered in surfactant–polymer composite systems, a branched functional polymer (PEM) was prepared from acrylamide, acrylic acid, polyethyleneimine, and a newly synthesized functional monomer (XN) to realize high efficient displacement of heavy oil in oil reservoir conditions. Proton nuclear magnetic resonance (1HNMR) analysis, elemental analysis, and Fourier-transform infrared spectroscopy verified the synthesis of PEM. The thickening performance, salt resistance, interfacial activity, ability of heavy oil viscosity reduction by emulsification, and core-flooding performance of PEM were systematically studied, with linear functional polymer PXM containing XN and conventional polymer HPAM as the control group. Under the synergistic effect of functional monomer XN and the branched structure, PEM demonstrated better thickening performance and more ideal heavy oil viscosity reduction compared with those of PXM and HPAM. The branched structure of PEM enhanced the interaction between molecules and the strength of the spatial network structure; notably, the thickening performance of PEM improved by 34.6% and 141.7%, and the salt resistance performance improved by 31.5% and 89.4%, compared with those of PXM and HPAM, respectively. Furthermore, PEM formed a relatively tight interfacial adsorption layer at the oil–water interface, leading to the reduction of interfacial tension. The heavy oil viscosity reduction rate of PEM (87.0%) was higher than that of PXM (79.5%). The emulsion formed was more stable than PXM, and the oil–water separation rate was slower. Owing to the better heavy oil viscosity reduction effect of PEM, the recovery factor of PEM (21.4%) was higher than that of PXM (17.2%) under the same polymer viscosity condition. These findings indicate that PEM and its functional mechanism are suitable for the enhancement of heavy oil recovery.