This paper considers a generic convex minimization template with affine constraints over a compact domain, which covers key semidefinite programming applications. The existing conditional gradient methods either do not apply to our template or are too slow in practice. To this end, we propose a new conditional gradient method, based on a unified treatment of smoothing and augmented Lagrangian frameworks. The proposed method maintains favorable properties of the classical conditional gradient method, such as cheap linear minimization oracle calls and sparse representation of the decision variable. We prove O(1/√k) convergence rate for our method in the objective residual and the feasibility gap. This rate is essentially the same as the state of the art CG-type methods for our problem template, but the proposed method is arguably superior in practice compared to existing methods in various applications.