Research suggests trust in experts and authorities are important correlates of compliance with public health measures during infectious disease outbreaks. Empirical evidence on the dynamics of reliance on scientists and public health authorities during the early phases of an epidemic outbreak is limited. We examine these processes during the COVID-19 outbreak in Italy by leveraging data from Twitter and two online surveys, including a survey experiment. We find that reliance on experts followed a curvilinear path. Both Twitter and survey data showed initial increases in information-seeking from expert sources in the three weeks after the detection of the first case. Consistent with these increases, knowledge about health information linked to COVID-19 and support for containment measures was widespread, and better knowledge was associated with stronger support for containment policies. Both knowledge and containment support were positively associated with trust in science and public health authorities. However, in the third week after the outbreak, we detected a slowdown in responsiveness to experts. These processes were corroborated with a survey experiment, which showed that those holding incorrect beliefs about COVID-19 gave no greater - or even lower - importance to information when its source was stated as coming from experts than when the source was unstated. Our results suggest weakened trust in public health authorities with prolonged exposure to the epidemic as a potential mechanism for this effect. Weakened responsiveness to expert sources may increase susceptibility to misinformation and our results call for efforts to sustain trust in adapting public health response.