In web search on debated topics, algorithmic and cognitive biases strongly influence how users consume and process information. Recent research has shown that this can lead to a search engine manipulation effect (SEME): when search result rankings are biased towards a particular viewpoint, users tend to adopt this favored viewpoint. To better understand the mechanisms underlying SEME, we present a pre-registered, 5 x 3 factorial user study investigating whether order effects (i.e., users adopting the viewpoint pertaining to higher-ranked documents) can cause SEME. For five different debated topics, we evaluated attitude change after exposing participants with mild pre-existing attitudes to search results that were overall viewpoint-balanced but reflected one of three levels of algorithmic ranking bias. We found that attitude change did not differ across levels of ranking bias and did not vary based on individual user differences. Our results thus suggest that order effects may not be an underlying mechanism of SEME. Exploratory analyses lend support to the presence of exposure effects (i.e., users adopting the majority viewpoint among the results they examine) as a contributing factor to users' attitude change. We discuss how our findings can inform the design of user bias mitigation strategies.