After the haunt does your turn still end in a newly discovered room that has an event or omen? If so this seems to make it very difficult to get away from monsters.
Yes, your movement will always end after you enter a newly discovered room with a symbol in it. This does mean you can have some claustrophobic haunts, but it is the way the game is played.
Actually, your movement will end if you draw a card for any reason, whether it was for revealing a new icon-bedecked room or for some other reason (such as opening the Puzzlebox).
From the rulebook (italics mine):
On your turn, you can move up to a number of spaces equal to your character's current Speed. You can perform actions (such as using an item or attacking) in the middle of your movement. However, whenever a game effect makes you draw a card for any reason, you must stop moving for the rest of the turn.
This rule does not change once the haunt has been revealed.