For graphs H and F with chromatic number χ(F ) = k, we call H strictly F -Turán-good (or (H, F ) strictly Turán-good) if the Turán graph T k−1 (n) is the unique F -free graph on n vertices containing the largest number of copies of H when n is large enough. Let F be a graph with chromatic number χ(F ) ≥ 3 and a color-critical edge and let P ℓ be a path with ℓ vertices. Gerbner and Palmer ( , arXiv:2006) showed that (P3, F ) is strictly Turán good if χ(H) ≥ 4 and they conjectured that (a) this result is true when χ(F ) = 3, and, moreover, (b) (P ℓ , K k ) is Turán-good for every pair of integers ℓ and k. In the present paper, we show that (H, F ) is strictly Turán-good when H is a bipartite graph with matching number ν(H) = ⌊ |V (H)| 2 ⌋ and χ(F ) = 3, as a corollary, this result confirms the conjecture (a); we also prove that (P ℓ , F ) is strictly Turán-good for 2 ≤ ℓ ≤ 6 and χ(F ) ≥ 4, this also confirms the conjecture (b) for 2 ≤ ℓ ≤ 6 and k ≥ 4.