How do I examine my iPad for malware?
If your device is not jailbroken the odds of malware on iOS are basically 0, at least for what you're describing. There is currently no way anything could be installed without going through the App Store. Even then, a malicious app could only access its own files in its own sandbox. So, no installed app on iOS could modify anything outside of it.

This is also why you won't find any app scanning your iPad for malware. A malware scanner would need access to all system files but no app on iOS could do that (except for Apple's own apps).

This sounds more like a problem made by the particular websites themselves.