The reason why you see Charles as the killer who offed Madame Fate? It's personal; he used to be an exhibit in her carnival before sent into an asylum!
In Dire Grove there are four people representing the four elements, wind, water, fire, and earth. In the end game you use the fifth element, the silver soul, in a puzzle. This is represented by the object mercury and the Detective is the person.
Several of the carnies that were shown getting killed in Madame Fate turn out to be alive, albeit subjected to torture or conscripted by the Big Bad, in Fate's Carnival. Given that their deaths were only shown in Fate's crystal ball, it's likely that its images were metaphorical in some cases, and that other visions (specifically, the acrobat twins' Disney Villain Death) actually foretold how they'd expire in the next game rather than that night.
But what about the other visions? Dr. Goodwell isn't devoured by his own snake, Dante isn't hung by his own tongue, Franco starves to death instead of eating a horse...
Franco's may have been a metaphor that the Master Detective didn't understand. The others may simply not have happened yet.
And what about: Twila, Lucy and Armaldo? They aren't shown. And Art didn't die of cigarettes, did he?
Possibly Art was drugged with tainted cigarettes to incapacitate him, then killed.
The backstory of Dire Grove. The banshee came into existence because of a virgin sacrifice - one which involved the girl being sealed in an ice cave until she died. It Makes Sense in Context, but if you give it any thought after you solve the mystery... no wonder she wanted revenge!
Who exactly drew the sketches in Emma's diary in Ravenhearst? Especially the one of Charles about to kill her with a hatchet? It could be that the detective herself was the one who drew most of them, but there's no indication that she should have known that particular detail. Artistic license?
Fridge Brilliance: Escape reveals that ghosts can alter the appearance of objects, so maybe the "sketches" were blank paper until the Master Detective started reading the diary and Emma made the drawings appear.
What use is it to track down and dispose of the cursed horn from Shadow Lake, if the rest of the demon skeleton is left lying around where anyone who snoops around the ruined prison could find it?
In Escape, it's revealed that Victor went back in time at the end of Return and set up everything that happens in Escape. Why didn't he, say, prevent the Master Detective from solving the mystery in the original Ravenhearst? Or stop her from being born?
Dire Grove has one that makes no sense at all. In the early parts of the game, the videos the detective finds clearly show that next to no one has any interest in Allison's theory about Dire Grove, and evidence crops up that the people leading the archaeological dig on the site think she's more or less out of her mind. Cut to the end, and she's giving her presentation to a packed auditorium; she's rewarded with thunderous applause and cheering. Are we supposed to believe that these people believe she went through all of that?
In Madame Fate and the first Ravenhearst game we see that the Master Detective has an in with the Queen and she collects a lot of evidence in her logs. And as a detective in the service of the Queen, any notes she collects can be proven as evidence and can vouch for Allison and the other three students. Her endorsement and reputation can lend legitimacy to Allison's talk.
The fact that a chunk of Britain verifiably did start to freeze for no meteorologically-explicable reason probably helped also.
Fate's Carnival contains several references to "the Ravenhearst family home" or to Alister's endeavors to restore the Ravenhearst family to power. But "Ravenhearst" was Emma's family name, and Alister hated the fact that Charles abandoned his mystical studies to pursue her. So wouldn't Alister want to restore the Dalimar family's power, and not give a damn about the mansion his son built for Emma?
So for Fate's Carnival, if the detective had Dalimar's diary the whole goddamn time why the fuck was it necessary to complete that final puzzle in the game when we could've just stabbed the freaking thing with the silver knife without wasting so much time going over a fake?