Infectious disease outbreaks are expected to grow exponentially in time when left unchecked. Containment measures such as lockdown and social distancing can drastically alter the growth dynamics of the outbreak. This is the case for the 2019-2020 COVID-19 outbreak, which is characterized by a power-law growth. Strikingly however, the power-law exponent is different across countries. Here I illustrate the relationship between these two extreme scenarios, exponential and power-law growth, based on the impact of superspreaders and lockdown strategies to contain the outbreak. The theory predicts a relationship between the power- law exponent and the time interval between the first case and lockdown that is validated by the observed COVID-19 data across different countries.