Summary: Given an edge-weighted graph \(G\) on \(n\) nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number \(k\) of crossings in a given drawing of \(G\). Our algorithm achieves a running time of \(O(2^k \cdot p(n+k))\), where \(p\) is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [C. Dahn et al., Lect. Notes Comput. Sci. 10979, 141–152 (2018; Zbl 1511.68200)] is restricted to embedded 1-planar graphs (i.e., at most one crossing per edge) and its dependency on \(k\) is of order \(3^k\). Finally, combining this with the fact that crossing number is fixed-parameter tractable with respect to itself, we see that Max-Cut is fixed-parameter tractable with respect to the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor-monotone-version of crossing number.