Aiming at solving the performance degradation caused by the covariance matrix mismatch in wideband beamforming for conformal arrays, a novel adaptive beamforming algorithm is proposed in this paper. In this algorithm, the interference-plus-noise covariance matrix is firstly reconstructed to solve the desired signal contamination problem. Then, a sparse reconstruction method is utilized to reduce the high computational cost and the requirement of sampling data. A novel cost function is formulated by the focusing matrix and singular value decomposition. Finally, the optimization problem is efficiently solved in a second-order cone programming framework. Simulation results using a cylindrical array demonstrate the effectiveness and robustness of the proposed algorithm and prove that this algorithm can achieve superior performance over the existing wideband beamforming methods for conformal arrays.