We present a projection-based iterative algorithm for computing general-form Tikhonov regularized solutions to the problem minx| Ax-b |2^2+lambda2| Lx |2^2, where the regularization matrix L is not the identity. Our algorithm is designed for the common case where lambda is not known a priori. It is based on a joint bidiagonalization algorithm and is appropriate for large-scale problems when it is computationally infeasible to transform the regularized problem to standard form. By considering the projected problem, we show how estimates of the corresponding optimal regularization parameter can be efficiently obtained. Numerical results illustrate the promise of our projection-based approach.