Policy induced decline of human mobility has been recognised to be effective in controlling the COVID-19 spread especially in the initial stage of the outbreak, although the relationship among mobility, policy implementation, and virus spread remains contentious. Coupling data of confirmed COVID-19 cases with Google mobility data in Australia, we present a state-level empirical study to: 1) inspect the temporal variation of COVID-19 spread and the change of mobility adherent to social restriction policies; 2) examine the extent that different types of mobility are associated with the COVID-19 spread in eight Australian states/territories; and 3) analyse the time-lag effect of mobility restriction on the COVID-19 spread. We find that social restriction policies implemented in the early stage of the pandemic controlled the COVID-19 spread effectively; the restriction of human mobility has a time-lag effect on growth rates, and the strength of the mobility-spread correlation increases up to seven days after policy implementation but decreases afterwards. The association between mobility and COVID-19 spread varies across space and time, and subjects to the types of mobility. Thus, it is important for governments to consider the degree to which lockdown conditions can be eased by accounting for this dynamic mobility-spread relationship.