This study considers 28 geomagnetic storms with Dst $\leq-50$ nT driven by high-speed streams (HSSs) and associated stream interaction regions (SIRs) during 2010-2017. Their impact on ionospheric horizontal and field-aligned currents (FACs) have been investigated using superposed epoch analysis of SuperMAG and AMPERE data, respectively. The zero epoch ($t_0$) was set to the onset of the storm main phase. Storms begin in the SIR with enhanced solar wind density and compressed southward oriented magnetic field. The integrated FAC and equivalent currents maximise 40 and 58 min after $t_0$, respectively, followed by a small peak in the middle of the main phase ($t_0$+4h), and a slightly larger peak just before the Dst minimum ($t_0$+5.3h). The currents are strongly driven by the solar wind, and the correlation between the Akasofu $\varepsilon$ and integrated FAC is $0.90$. The number of substorm onsets maximises near $t_0$. The storms were also separated into two groups based on the solar wind dynamic pressure p_dyn in the vicinity of the SIR. High p_dyn storms reach solar wind velocity maxima earlier and have shorter lead times from the HSS arrival to storm onset compared with low p_dyn events. The high p_dyn events also have sudden storm commencements, stronger solar wind driving and ionospheric response at $t_0$, and are primarily responsible for the first peak in the currents after $t_0$. After $t_0+2$ days, the currents and number of substorm onsets become higher for low compared with high p_dyn events, which may be related to higher solar wind speed.