Generalized Turán results for disjoint copies of degenerate graphs

The generalized Turán number ex(n, H, F) denotes the maximum number of copies of H in an n-vertex F-free graph. For an integer t ≥ 1, let $tF$ be the vertex-disjoint union of t copies of F. Gerbner, Methuku, and Vizer (2019) established an asymptotically sharp bound for ex(n,Kᵣ,(t+1)K2,b). We extend their results in two directions by considering forbidden graphs (t+1)Ka,b and (t+1)C₂ₖ and establish more precise matching upper and lower bounds of the same order of magnitude.