Context. Extragalactic jets are judged to harbor dynamically important, organized magnetic fields that presumably aid in the collimation of the relativistic jet flows. Aims. We here explore the morphology of AGN jets pervaded by helical field and flow topologies by means of grid-adaptive, highresolution numerical simulations. We concentrate on morphological features of the bow shock and the jet beam behind the Mach disk, for various jet Lorentz factors and magnetic field helicities. We investigate the influence of helical magnetic fields on jet beam propagation in an overdense external medium. We adopt a special relativistic magnetohydrodynamic (MHD) viewpoint on the shockdominated AGN jet evolution. Due to the adaptive mesh refinement (AMR), we can concentrate on the long-term evolution of kinetic energy-dominated jets, with beam-averaged Lorentz factor Γ 7, as they penetrate denser clouds. These jets have near-equipartition magnetic fields (with the thermal energy) and radially varying Γ(R) profiles within the jet radius R < R j maximally reaching Γ ∼ 22. Methods. We used the AMRVAC code, with a novel hybrid block-based AMR strategy, to compute ideal plasma dynamics in special relativity. We combined this with a robust second-order shock-capturing scheme and a diffusive approach to controlling magnetic monopole errors. Results. We find that the propagation speed of the bow shock systematically exceeds the value expected from estimates using beamaverage parameters, in accordance with the centrally-peaked Γ(R) variation. The helicity of the beam magnetic field is effectively transported down the beam, with compression zones between the diagonal internal cross-shocks showing stronger toroidal field regions. In comparison with equivalent low-relativistic jets (Γ 1.15), which get surrounded by cocoons with vortical backflows filled by mainly toroidal field, the high speed jets only demonstrate localized, strong toroidal field zones within the backflow vortical structures. These structures are ring-like due to our axisymmetry assumption and may further cascade to a smaller scale in 3D. We find evidence of a more poloidal, straight field layer, compressed between jet beam and backflows. This layer decreases the destabilizing influence of the backflow on the jet beam. In all cases, the jet beam contains rich cross-shock patterns, across which part of the kinetic energy gets transfered. For the high-speed reference jet considered here, significant jet deceleration only occurs beyond distances exceeding O(100R j ), as the axial flow can reaccelerate downstream to the internal cross shocks. This reacceleration is magnetically aided by field compression across the internal shocks that pinch the flow.