Generalized Turán results for disjoint copies of degenerate graphs

The generalized Turán number $\mathrm{ex}(n, H, F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. For an integer $t \geq 1$, let $tF$ be the vertex-disjoint union of $t$ copies of $F$. Gerbner, Methuku, and Vizer (2019) established an asymptotically sharp bound for $\mathrm{ex}(n,K_r,(t+1)K_{2,b})$. We extend their results in two directions by considering forbidden graphs $(t+1)K_{a,b}$ and $(t+1)C_{2k}$ and establish more precise matching upper and lower bounds of the same order of magnitude.