In this article, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg $L$ -functions. One of the main new inputs is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new subconvexity bound for Rankin-Selberg $L$ -functions in the level aspect. Moreover, infinitely many Rankin-Selberg $L$ -functions having at most eight nontrivial real zeros are produced, and some new nontrivial estimates for the analytic rank of the family studied are obtained