The time to give up on a hand in Hold 'Em is either before the flop or on the flop. If you still have an active hand by the turn, you're usually in for the duration.

The betting rounds for the last two cards are at double the bet size of the first two betting rounds. By the time you reach this point, your hand and the likely holdings of your opponents should be fairly well defined. At the first of these last betting rounds, six of your seven cards have been exposed. You only have one more card coming. At this point you generally either have the probable best hand, or you have a draw to the probable best hand.

Most of the time, if your hand was good enough to stay for the turn card, then it's good enough to stay for the river card. Not always though. An example of when it might not be right to stay past the turn card is when the pot was offering you enough odds to draw one card for an inside straight, but when the bet doubles on the turn round, the pot is not twice the size and you're no longer getting good enough odds to call.

Generally, if your draw on the flop was strong enough for an automatic call, then it'll still be good enough to call on the turn. If your call on the flop depended on a close analysis of pot odds, then it's probably not good enough to call the larger turn bet.