The scene where Rasputin invades Anya's dream to make her fling herself over the edge of the boat? Eeep! The music alone is scary!
The tail-end where the dream turns into a nightmare was so scary it was actually edited out of some releases of the film.
The death of Rasputin and his being disintegrated into dust. Also Rasputin making his Deal with the Devil in the opening sequence.
And briefly having his skin torn off during the above-mentioned scene so that he briefly became a skeleton!
He was stripped all the way to the bone; that means his skin, muscles, and organs were ripped from his skeletal frame. Makes one hope that the Dark Forces were kind enough to make it painless, otherwise... Ugh.
Rasputin trying to make Anya fall overboard off the ship, into the icy water below.
Young Anastasia: Let me go! Please!
Rasputin: You'll never escape me, child, never!
Then there's the spin-off movie Bartok The Magnificent. Especially since the main villainess has a potion that turns whoever drinks it into what they were like on the inside; when Ludmilla drinks it, she turns into a dragon of some kind. It's not a pleasant sight.
There was a PC game where you played as Pooka, exploring Russia and helping out Anasastia and co. or random strangers; however, for a good chunk of time you're usually alone. This can be slightly off-putting to a young child, but by itself it's not too bad...until you search the wrong spot and trigger an unwelcome visit from a minion who teleports you to the Underworld. All you have to do to escape is win one of the minigames, but oftentimes there's absolutely no indication of what's safe and what's not. Bartok will warn you that there's a trap in the room you've entered...sometimes.
When Rasputin corners the empress and Anastasia on the frozen Neva River—first he simply tries to kill a terrified little girl, while her grandmother can only struggle with him in horror; then, when the ice breaks beneath them, he tries to pull Anastasia in with him until eventually they break free, allowing him to drown (but not die)...