Conspiracy theory =
thinking there is something dodgy going on without any tangible proof, or that something exists or doesn't exist, again with no proof, but they believe it because they don't trust the Government / Big Whoever and someone on the internet let them in on the secret so it must be true. eg: the moon is a projection or that birds aren't real (and if they're not, what have I been eating on Christmas Day all this time?).
That's fair enough as one definition; but again, so many potentially plausible things that 'really don't seem right' will start off with no tangible proof. Many successfully-convicted crimes initially begin with just a hunch or suspicion that all may not seem what it should be.
People may well not completely trust the government - to be honest, I'd be surprised if anybody does, including many people who work in government. This can come from a place of paranoia or healthy scepticism - possibly going on historical events/appearances (e.g. the deaths of David Kelly and Robin Cook) - or anywhere in between.
The thing is that, if the government were conspiring to do something big and nefarious, how would the ordinary person ever be able to gather any meaningful evidence to satisfy themselves apart from looking at the available info and personally weighing it all up? If somebody did stumble across some damning incontrovertible evidence that implicated extremely powerful people, do you believe that those high-up people would sigh and accept that they'd been caught and take their serious punishment - when they could very easily discredit the 'fantasist' who discovered it, also warning off anybody else who might get too close, and use their power to protect themselves?
Take the conspiracy theory that Prince Philip killed Princess Diana. I think there were several far-from-unreasonable circumstances that may have led people to wonder - not least the proclamations of Diana herself that 'they' were going to bump her off in a car accident. Now, the official investigation took due course and the finding was that it was basically just a tragic accident. However, just supposing that PP had arranged for the murder of Diana - bearing in mind that any person could decide to commit any crime, however surprising it may be - do you honestly, genuinely believe that this would have ever been admitted to the public?