One of the largest nationwide bursts of the first COVID-19 outbreak occurred in Spain, where infection expanded in densely populated areas through March 2020. We analyse the cumulative growth curves of reported cases and deaths in all Spain and two highly populated regions, Madrid and Catalonia, identifying changes and sudden shifts in their exponential growth rate through segmented Poisson regressions. We associate these breakpoints with a timeline of key events and containment measures, and data on policy stringency and citizen mobility. Results were largely consistent for infections and deaths in all territories, showing four major shifts involving 19-71% reductions in growth rates originating from infections before 3 March and on 5-8, 10-12 and 14-18 March, but no identifiable effect of the strengthened lockdown of 29-30 March. Changes in stringency and mobility were only associated to the latter two shifts, evidencing an early deceleration in COVID-19 spread associated to personal hygiene and social distancing recommendations, followed by a stronger decrease when lockdown was enforced, leading to the contention of the outbreak by mid-April. This highlights the importance of combining public health communication strategies and hard confinement measures to contain epidemics.