Damage is applied to turtle shield post-mitigation. Therefore, buying defensive items can drastically increase your survivability.
Udyr is one of the best junglers in the game. Taking advantage of this can give your team a large XP advantage and map control.
Udyr players will often start in the jungle and only appear to gank in lanes. Try warding key locations to keep track of his movement.
Once the initial shield effect has faded, Turtle Stance is weak against other champions. Press the attack to force him to switch stances and prevent him from regenerating health.
Some Udyr players buy tank items to become very durable. If they do, focus their weaker teammates instead.