I'm thinking specifically about Lost Legacy. If I get them to exile 4 cards ie. Saheeli Rai, and allow them to draw 4 on my turn does that mean they can draw on their turn and do not have to discard until the end of their next turn?
Yes, they still get to draw and do not have to discard down to 7 until their cleanup step.
402.2. Each player has a maximum hand size, which is normally seven cards. A player may have any number of cards in his or her hand, but as part of his or her cleanup step, the player must discard excess cards down to the maximum hand size.