Honestly, addressing just one of these issues would put Udyr in a good place. Two would probably break him.
**1. His built-in survivability and damage are both insufficient after level 5.**
Having to switch between stances only allows him to have one of survivability, damage, or mobility at any given time, and he doesn't get an ultimate, so his power falls off a cliff in comparison to other champions when they hit 6.
If you're in Bear Stance for mobility, you don't have damage or survivability.
If you're in Turtle Stance for survivability, you don't have any damage or mobility.
If you're in Tiger Stance for damage, you don't have any mobility or survivability.
This doesn't work, because he plays like a juggernaut: he runs at his opponent and tries to kill them quickly if he manages to get on top of them. Actual juggernauts usually get all of these things at the same time, instead of just one at a time, and they often have either innately high damage or survivability in their kit, so they're able to focus on whatever is lower. His baseline kit doesn't provide enough survivability or damage, especially given that he only one of them is active at a time.
Tiger Stance relies on its AD scaling for most of its damage, and Turtle Stance's survivability is weak. The shield has no scaling and the healing is low, especially when you consider that he's giving up both mobility and damage in order to use it. If you compare it to something like Darius's Q or Nasus's passive life steal, it's just an incredibly bad ability.
One solution to this would be to make Turtle Stance a better ability that has more synergy with non-tank builds, possibly by giving it some AD scaling. It's usually not worth leveling it past rank 1, because the healing doesn't scale at all and the shield is fairly weak.
**2. His build paths are terrible and inefficient.**
He wants to build like a juggernaut; in order to perform his function, every item he builds should have some combination of survivability, damage, and mobility, ideally all three. There's a big problem with this: there is no juggernaut jungle item, so he has to choose between either having no damage or no survivability.
If he builds Cinderhulk, he runs into problems securing kills (assuming he can even get on top of his target) and is generally ineffective until he completes his second item, but it's much safer. If he builds Warrior, he can kill targets he gets on top of and duel better, but the risk becomes too high. If he gets CC'd during a gank or the enemy jungle gets the drop on him post-6, they can likely one shot him, and and ganking burst champions like Syndra becomes suicide.
You can somewhat work around this by rushing Trinity Force, instead of upgrading your jungle item. In most games, this is the best option, but it still suffers from the problem that you don't get your first item power spike until 16-17 minutes, while other jungles will get theirs around 10-11 minutes. This puts you at disadvantage unless you're pretty far ahead.
This doesn't take into account that Udyr has problems farming the jungle and pushing lanes, because he has no AoE when he goes Tiger. This makes his clear of krugs and raptors very slow, and he takes a lot of damage, especially early, which impacts his ability to gank. This leaves you with three options:
1. Max Phoenix instead of Tiger. This is usually not a good idea. Phoenix is almost always inferior to Tiger and has no scaling with the items he builds, so he falls off even harder than usual and has less dueling potential.
2. Rush Tiamat. Unfortunately, this sets him back 1325 gold and further delays his Trinity Force power spike. You're almost paying as much as you would to upgrade your jungle item, and you have the same problems as if you had built Warrior, with less of the power.
3. Do neither, and hope you can make up the gold difference with kills. This is insanely risky, because Udyr's ganks are poor and rely on the opponent being unaware and over extended.
There's too much inefficiency here, which prevents him from being competitive with other junglers. Assuming there's no chance of a juggernaut jungle item being added, a possible solution to at least help his clear would be to add an AoE that only hits minions/monsters to the Tiger Stance proc.
**3. The movement speed soft cap hinders his mobility too much.**
Movement speed over 415 is reduced by 20% and movement speed over 490 is reduced by 50%.
Udyr relies on regular small bursts of speed to semi-constantly move 'sort of fast,' but he's hovering around 500 movement speed for most of the game. The numbers look nice on paper, but he's constantly feeling the full 20% soft cap. It's extremely noticeable when playing him; even with hundreds of games on Udyr, I still sometimes double check that I remembered to buy boots, because I'm not moving as fast as I feel like I should be.
Other champions who rely on movement speed, like Rammus and Nunu, get a massive burst of movement speed and exceed the cap by so much that they don't really notice it. This type of mobility is also more useful in 90% of scenarios, especially when ganking, because it allows them to minimize the time their opponent has to notice them and react.
Touching his mobility is tricky, because his consistent movement speed can be overbearing against teams with no CC (however rare they are), but flattening either the movement speed bonus or duration on Bear Stance so they no longer scale with rank would likely help.