In this paper, we introduce Kirk-multistep and Kirk-multistep-SP iterative schemes and prove their strong convergences and stabilities for contractive-type operators in normed linear spaces. By taking numerical examples, we compare the convergence speed of our schemes (Kirk-multistep-SP iterative schemes) with the others (Kirk-SP, Kirk-Noor, Kirk-Ishikawa, Kirk-Mann and Kirk iterative schemes) for this class of operators. Our results generalize and extend most convergence and stability results in the literature. MSC: 47H09; 47H10