If popular online platforms systematically expose their users to partisan and unreliable news, they could potentially contribute to societal issues like rising political polarization. This concern is central to the echo chamber and filter bubble debates, which critique the roles that user choice and algorithmic curation play in guiding users to different online information sources. These roles can be measured in terms of exposure, the URLs seen while using an online platform, and engagement, the URLs selected while on that platform or browsing the web more generally. However, due to the challenges of obtaining ecologically valid exposure data--what real users saw during their regular platform use--studies in this vein often only examine engagement data, or estimate exposure via simulated behavior or inference. Despite their centrality to the contemporary information ecosystem, few such studies have focused on web search, and even fewer have examined both exposure and engagement on any platform. To address these gaps, we conducted a two-wave study pairing surveys with ecologically valid measures of exposure and engagement on Google Search during the 2018 and 2020 US elections. We found that participants' partisan identification had a small and inconsistent relationship with the amount of partisan and unreliable news they were exposed to on Google Search, a more consistent relationship with the search results they chose to follow, and the most consistent relationship with their overall engagement. That is, compared to the news sources our participants were exposed to on Google Search, we found more identity-congruent and unreliable news sources in their engagement choices, both within Google Search and overall. These results suggest that exposure and engagement with partisan or unreliable news on Google Search are not primarily driven by algorithmic curation, but by users' own choices.