We prove a version of Bourgain’s projection theorem for parametrized families $C^2$ maps, which refines the original statement even in linear case by requiring non-concentration only at single natural scale. As one application, we show that if $A$ is Borel set Hausdorff dimension close to $1$ $\mathbb{R}^2$ or $3/2$ $\mathbb{R}^3$, then $y\in A$ outside very sparse set, pinned distance ${|x-y|:...