Democracy isn't just about who wins; it's about how they win.
In which case we have rarely lived in a Democracy. Every election contains subterfuge, twisting of facts, etc. There have also been claims of rigging, overspending, etc in the past but it doesn't usually mean we get to change the result of the election. The Project Fear tactics of the Remain Campaign was outrageous too but I suspect if they'd have won, the Leavers would be expected to abide by the result.
I can be confident in the way I voted because I didn't just listen to the news (which tended to be biased towards Remain) or what the Leave Campaign were saying. I have watched for years what is happening, saw how Greece were treated, thought about the burgeoning bureaucracy, the length of time it takes to make progress, etc and made my choice. I suspect a vast amount of people did the same. I also think the Remainers were so confident they would win they didn't enter the polling booth.
In a way, I would be happy to have a new referendum because I don't think it would necessarily be as clear cut as you think it would be. A lot of leavers do feel intimidated to admit they voted leave and would do again so in the Polls don't always admit their leaning. I just don't think we have time to do this because we are on the wire. We can't carry on in limbo forever.
Anyway, we could debate this ad infinitum but it won't get us anywhere, we are never going to agree. We just have to wait for Parliament to decide what they are going to do. Whichever side wins, there are going to be some very sore losers.