Birationally rigid complete intersections of high codimension

We prove that a Fano complete intersection of codimension $k$ and index $1$ in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ with at most multi-quadratic singularities is birationally superrigid. The codimension of the complement of the set of birationally superrigid complete intersections in the natural moduli space is shown to be at least $(M-5k)(M-6k)/2$. The proof is based on the technique of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities.