What do you mean exactly ? Other side like should be on the right and this is on the left ?
Only thing I think off is related to how your other website do with CSS or JS. Maybe there is some specific styling or processing that will modify the one from the html produced with rmarkdown.
html_document() will produce a full HTML document with
<body> and will rely on some resources like specific CSS and JS library depending on the feature you are using (example, it uses bootstrap by default).
When you include in another website, this could all conflict with your website style, resources and rules. Usually not easy to include a full HTML in another place without doing an
html_fragment() format too that will only produce a
<body> part, but some feature will be missing.
Anyhow, not sure how to solve this except trying to isolate the possible CSS rules that are impacting