Countries across the world responded to the COVID-19 pandemic with what might well be the set of biggest state-led mobility and activity restrictions in the history of mankind. But how effective were these measures across countries? Compared to multiple recent studies that document an association between such restrictions and the control of the contagion, we use an instrumental variable approach to estimate the causal effect of these restrictions on mobility and the growth rate of confirmed cases and deaths attributed to COVID-19. Using the level of stringency of the restrictions in the rest of the world to predict the level of stringency of the restriction in a country, we show that while stricter contemporaneous measures affected mobility, stringency in seven to fourteen days prior mattered for containing the contagion. Heterogeneity analyses reveal that even though the restrictions reduced mobility more in relatively less-developed countries, the causal effect of a reduction in mobility was higher in more developed countries. We propose several explanations. Our results highlight that to combat the COVID-19 pandemic effectively, mobility and activity restrictions must be complemented with other health and information measures.