Have you checked the Developer Tools (F12) → Console for any errors like mixed content, etc.?
Before moving to Cloudflare, was your Website working over HTTPS connection?
What kind of a content have you got?
Is there a lot of traffic / visitors?
Any images or videos, or embeded iframes?
Is it a WordPress website or some webshop?
Might mean your server is generating a page, therefore upon refresh it gets cached after some particular time and is kept for some time like that n your web browser until it’s expired at the origin host and therefore the server has to re-generate the cached HTML document to serve it again.
However, I might be wrong about it because I don’t have the example URL neither know the domain name to test this out. Therefore, you’ve mentioned you are on a ticket with Business plan, I believe they’d figure and sort this out for you in the meantime.