H-Darrieus vertical-axis wind turbines (VAWTs) capture wind regardless of its direction and operate effectively even in challenging and turbulent wind conditions. As a result, the blades operate under erratic and intricate aerodynamic loads, which cause them to bend. The performance of the H-Darrieus rotor will therefore be impacted by the blade’s deflection. This study aims at investigating the dynamic aerostructure influence on a novel convex-bladed H-Darrieus geometry. The results are compared to a straight-bladed baseline rotor. To do so, a two-way fluid–structure interaction (FSI)-coupled approach is performed to accurately address this issue. This approach allows for the simultaneous resolution of the fluid flow around the rotor and the mechanical structure responses inside the blades. The turbulent flows are resolved using the k-ω-SST model together with the URANS equations through computational fluid dynamics (CFD), while the structural deflections of the blades are assessed using finite element analysis (FEA). The results show that the performance of both H-Darrieus turbines decreases with increasing deformation. In addition, the study found that the carbon fiber composite (M1) material has the least deformation in the convex and straight blades, with values of 9.1 mm and 20.331 mm, respectively. The glass-fiber-reinforced epoxy composite (M3) material shows the most significant deflection across both types, with displacements of 32.50 mm and 73.78 mm for the straight blade and 19.02 mm and 43.03 mm for the convex blade. This study also reveals that the straight blade has a peak displacement of 73.785 mm when using the M3 material at TSR = 3, while the convex blade has a minimum displacement of 20.331 mm when using the M1 material, highlighting the varying performance characteristics of the materials. The maximum stress observed occurs in the straight blade, registering at 324.1 MPa with TSR = 3, which aligns closely with the peak displacement values, particularly for the aluminum alloy material (M2). In contrast, the convex blade made from the first material (M1) exhibits the lowest stress levels among the tested configurations.