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Dimension 4 is famously wild, and classifying smooth, closed 4-manifolds is very difficult. Due to the failure of \textit{Whitney’s trick} in dimension 4, understanding embeddings of surfaces in 4-manifolds becomes crucial. For a 4-manifold $X$, and any class $a \in H_{2}(X,\mathbb{Z})$, there always exists a smoothly embedded surface representing this class. The \textit{minimum genus function} of $X$ assigns to each such class $a$ the minimal genus among all embedded surfaces that represent it: \[ g_{X}(a) = \min_{\Sigma} \{ g(\Sigma) \mid [\Sigma] = a \} \] This smooth invariant can, in principle, detect exotic smooth structures, but computing it is difficult and has been fully carried out only in a few cases. In ongoing work, I establish new upper bounds for various 4-manifolds, most effectively demonstrated in the case of connected sums of $S^2 \times S^2$, $\mathbb{CP}^2$, and $\overline{\mathbb{CP}^2}$. The techniques are purely topological and build upon previous work with M.~Marengon, in which we generalize the \textit{Norman--Suzuki trick}. The main idea is to simultaneously remove many singularities of immersed surfaces by employing more intricate tubing techniques.