We investigate the distortion incurred when embedding the union of $k$ Euclidean sets into a Euclidean space. The main theorem shows that, regardless of the total number of points, any union of $k$ Euclidean sets can be embedded with distortion at most $O\big(k^{1.128}\log k\big)$. This improves the previous bound $2^{O(k)}$, which was obtained by iterating the result of Makarychev and Makarychev (2016) for the special case $k=2$, by an exponential factor. Moreover, under the additional assumption that each Euclidean set remains Euclidean after adjoining any two extra points, we achieve a distortion bound of $O(\log k)$, which is asymptotically tight and thus generalizes Bourgain’s 1985 theorem. Along the way we also prove that the overall distortion is bounded by $O(\log k)$ times the maximum Lipschitz extension constant of the individual Euclidean sets.\n\nReview: The paper delivers a substantial leap from exponential to near‑linear distortion bounds in metric embedding theory, and under a mild geometric condition recovers the classic logarithmic bound, offering a powerful extension of Bourgain’s theorem.