Towards linear growth for biprojective hypersurfaces of small dimension

After removing suitable thin subsets, we count rational points of bounded height on some hypersurfaces of small bidegree in ${\mathbb{P}}^3\times {\mathbb{P}}^3$ and in ${\mathbb{P}}^3\times {\mathbb{P}}^2$. One of the ingredients is a new estimate for the number of rational points of bounded height on diagonal quadratic surfaces which improves as the coefficients grow.