Politicians often possess high media visibility and significant followings on social media, enabling them to influence public discourse and media narratives. Understanding their role in spreading misinformation online is crucial to mitigating societal harm. This study analyzed data from X (formerly Twitter) to examine the misinformation-sharing behavior of politicians and corresponding public engagement in Germany, Italy, the UK, and the US during 2020-2021. The analysis focused on three levels of political hierarchy: national executive, national legislative, and regional executive. The findings highlight significant differences in misinformation-sharing patterns, with politicians in Italy and the US posting substantially more misinformation, both in absolute and relative terms. Public engagement with misinformation also showed notable cross-country variations, considering both cumulative interactions and average engagement per post. During the COVID-19 pandemic, Italy emerged as a hotspot for crisis-related misinformation sharing and engagement. Across all countries, national and regional executive politicians exhibited minimal misinformation-sharing, while far-right and conservative legislators were the primary sources of such content, aligning with existing evidence.