  We describe a new method to obtain weak subconvexity bounds for $L$-functions
with mild hypotheses on the size of the Dirichlet coefficients. We verify these
hypotheses for all automorphic $L$-functions and (with mild restrictions) the
Rankin-Selberg $L$-functions attached to two automorphic representations. The
proof relies on a new unconditional log-free zero density estimate for
Rankin-Selberg $L$-functions.
