Can I dredge off a Bazaar if my opponent has a Possessed Portal in play? What about off my draw phase?
Dredge is a replacement effect. It replaces any draw with putting N cards from your library into your graveyard to return the card to your hand.
[card]possessed portal[/card] also has a replcement effect that replaces card drawing. Since you would be the 'controller' of the affected card draw, you may choose which replacement effect to use. If you use the dredge, then there is no card draw for possessed portal to replace. If you use the portal, then there is no draw for dredge to replace.
419.9a If two or more replacement or prevention effects are attempting to modify the way an event affects an object or player, the affected object's controller (or its owner if it has no controller) or the affected player chooses one to apply. Then the other effect applies if it is still appropriate. If one or more of the applicable replacement effects is a self-replacement effect (see rule 419.6d), that effect is applied before any other replacement effects. If two or more players have to make these choices at the same time, choices are made in APNAP order (see rule 103.4).
Example: Two permanents are in play. One is an enchantment that reads "If a card would be put into a graveyard, instead remove it from the game," and the other is a creature that reads "If [this creature] would be put into a graveyard from play, instead shuffle it into its owner's library." The controller of the creature that would be destroyed decides which replacement to apply first; the other does nothing.