Liquid-like at rest, dense suspensions of hard particles can undergo striking transformations in behaviour when agitated or sheared(1). These phenomena include solidification during rapid impact(2,3), as well as strong shear thickening characterized by discontinuous, orders-of-magnitude increases in suspension viscosity(4-8). Much of this highly non-Newtonian behaviour has recently been interpreted within the framework of a jamming transition. However, although jamming indeed induces solid-like rigidity(9-11), even a strongly shear-thickened state still flows and thus cannot be fully jammed(12,13). Furthermore, although suspensions are incompressible, the onset of rigidity in the standard jamming scenario requires an increase in particle density(9,10,14). Finally, whereas shear thickening occurs in the steady state, impact-induced solidification is transient(2,15-17). As a result, it has remained unclear how these dense suspension phenomena are related and how they are connected to jamming. Here we resolve this by systematically exploring both the steady-state and transient regimes with the same experimental system. We demonstrate that a fully jammed, solid-like state can be reached without compression and instead purely with shear, as recently proposed for dry granular systems(18,19). This state is created by transient shear-jamming fronts, which we track directly. We also show that shear stress, rather than shear rate, is the key control parameter. From these findings we map out a state diagram with particle density and shear stress as variables. We identify discontinuous shear thickening with a marginally jammed regime just below the onset of full, solid-like jamming(20). This state diagram provides a unifying framework, compatible with prior experimental and simulation results on dense suspensions, that connects steady-state and transient behaviour in terms of a dynamic shear-jamming process.