Yes. Though those cards end up in another
zone, they count as being drawn by the player, and will count against the one
spell that can be drawn per turn. For example, if a player draws a
site at the start of their turn and then plays
Morgana le Fay, Morgana will only
draw 1 spell into her
hand, resulting in a total of only 1 spell drawn by that player that turn. See the
Only It Can Cast entry for more information.