Exchange bias, as one of the core working principles of spintronics, is widely used in the development of high-sensitivity, high-stability, and high-density spintronic devices. As an essential phenomenon in exchange-biased systems, asymmetric magnetization reversal has been extensively studied in antiferromagnetic/ferromagnetic systems. However, this phenomenon has not yet received an adequate theoretical explanation in hard magnetic (HM)/soft magnetic (SM) systems. This work uses micromagnetic simulations to investigate the exchange bias and asymmetric magnetization reversal in the FeCo layer of $L{1}_{0}$ FePt (HM)/FeCo (SM) bilayer thin film with in-plane anisotropy. It is found that different thicknesses of the FeCo layer and varying angles between the applied magnetic field (${\mathbit{H}}_{\mathrm{appl}}$) and the magnetic easy-axis direction can change the magnetization reversal mechanism of the FeCo layer, thereby influencing the exchange bias and the asymmetry of magnetization reversal. An asymmetry in the hysteresis loop is observed when ${\mathbit{H}}_{\mathrm{appl}}$ is in the angular range near the direction of the easy axis, and the increase of the FeCo layer thickness expands the angular range in which asymmetry occurs. The interaction between uniaxial anisotropy and unidirectional anisotropy influences the magnetization reversal, causing one branch of the hysteresis loop to be dominated by coherent rotation of the magnetic moments. In contrast, the main magnetization reversal mechanism of the other branch is the nucleation of magnetic domains and further propagation of magnetic domain walls, thus forming an asymmetric hysteresis loop. The exchange bias and asymmetry of magnetization reversal can be controlled by changing the FeCo layer thickness and the angle between ${\mathbit{H}}_{\mathrm{appl}}$ and the easy axis. These understandings will provide additional ideas for research on magnetic devices.