A conforming partition of a rectilinear n-gon P is a partition of P into rectangles without using Steiner points (i.e., all corners of all rectangles must lie on the boundary of P). The stabbing number of such a partition is the maximum number of rectangles intersected by an axis-aligned segment lying in the interior of P. In this paper, we examine the problem of computing conforming partitions with low stabbing number. We show that computing a conforming partition with stabbing number at most 4 is \( \mathcal {N\!P}\) -hard, which strengthens a previously known hardness result [Durocher & Mehrabi, Theor. Comput. Sci. 689: 157–168 (2017)] and eliminates the possibility for fixed-parameter-tractable algorithms parameterized by the stabbing number unless \( \mathcal {P}= \mathcal {N\!P}\) . In contrast, we give (i) an \( O( n \log n ) \) -time algorithm to decide whether a conforming partition with stabbing number 2 exists, (ii) a fixed-parameter-tractable algorithm parameterized by both the stabbing number and treewidth of the pixelation of the polygon, and (iii) a fixed-parameter-tractable algorithm parameterized by the stabbing number for polygons without holes in general position.