By the well-known result of Yood, every strictly transitive algebra of operators on a Banach space is WOT-dense. This motivated us to investigate the relationships between SOT and WOT largeness of sets of operators and the transitivity behavior of them. We show that, to obtain Yood's result, strict transitivity may not be replaced by the weaker condition of hypertransitivity. We prove that, for a wide class of topological vector spaces, every SOT-dense set of operators is hypertransitive. The general form of SOT-dense sets that are not strictly transitive is presented. We also describe the form of WOT-dense sets that are not hypertransitive. It is shown that a set is hypertransitive if and only its SOT-closure is hypertransitive. We introduce strong topological transitivity and we show that every separable infinite-dimensional Hilbert space supports an invertible topologically transitive operator that is not strongly topologically transitive.