In many card games, participants are only required to hold a few cards at once, Poker being one of them. That’s normally not too difficult, even for a child or someone with small hands. But in bridge, and other similar games, you need to hold a hand of thirteen cards. The wider the cards, the harder it is to hold them all in such a way that you can see them and pick one out at will. The reduced width of bridge cards (about a quarter of an inch narrower than poker cards) helps people to hold them easier. By being easier to hold, there is a reduced risk that someone will drop and accidentally reveal a card.
Bridge tournaments are very competitive, where dropping a single card would be much worse than exposing a single card from your hand in poker.
The narrower size of Bridge size cards became more commonplace in poker rooms, as holdem involves "peeking" at only the very corner of your card. The wider Poker size requires lifting one (or more in the case of Omaha) even higher off the table where a conspiring friend of an opponent could glimpse your cards and signal what you are holding. Not an issue in most home games (were all friends at a home game), so many people still choose to use Poker size for poker. However, most casinos only use Poker size now for Blackjack and other card-using table games, so have adapted to using Bridge size for poker for an "authentic" casino experience.