In this paper, we address a discrete operator based on the composition of Ismail–May operator and Szász–Mirakjan operator and study its approximation properties for various classes of continuous functions using Taylor's formula. We estimate its moments and further give direct theorems employing Peetre's K‐functional and the modulus of continuity. Furthermore, Voronovskaja‐type asymptotic theorems are given. Additionally, we provide further compositions and give a comparison among their approximation properties.