Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. In order to reduce this sensitivity, typically, regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term and are popularly used to solve the ill-posed problems. Recently, the use of a $p$-norm to measure the fidelity term, and a $q$-norm to measure the regularization term, has received considerable attention. This paper presents a new efficient approach for the solution of the $p$-norm and $q$-norm minimization model of large discrete ill-posed problems, based on the majorization-minimization framework and the Golub-Kahan Lanczos bidiagonalization process, by using the discrepancy principle to choose the regularization parameters, called Majorization-Minimization Generalized Golub-Kahan Lanczos bidiagonalization regularization method (MM-GKL). The proof of the convergence analysis is provided. Numerical experiments illustrate that the proposed new method is more effective and less computational cost than the existing methods. Computed image restoration examples illustrate that it suffices to carry out less computational cost to achieve higher quality restorations. The combination of a low iteration count and a less computational cost requirement makes the proposed method attractive.