What it sounds like is that your browser isn't rendering CSS, JS and other web components correctly used for the website. This to me sounds like a cache issue, an add-on issue, or some setting in the browser. To verify if this is the case, run
portable Firefox (it's stand alone) once and go to the problematic website. If that works then you have a browser issue with Opera and we can rule out the computer, modem or router. So then we need a HAR file from Opera. A HAR file will allow me to see what exactly is going on. Follow
these instructions. Scroll down to where it tells you how to generate a HAR file in Opera. DO NOT login to the website while the HAR file is being generated. You just want to record the activity when you go to the problematic website. Once the HAR file is generated, Zip the resulting HAR file and upload that here.
Something else, are you using the latest version of Opera? No proxy? I believe Opera has their own proxy and this probably could mess things up. Check if there's a proxy in Opera turned on.
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Just for the heck of it, try the portable version of Opera as well as Firefox. Don't run both Opera's at the same time. Just launch this portable version (it's stand alone), go to the website in question and tell me if the site renders as it should.
Opera portable | USB browser | Free download | Opera
Only generate the HAR file in your problematic browser.