With a lot of pain.
Either procedural animation, or some clever IK manipulations.
Honestly I don’t really know how to do that, but one idea could be to have a dog controller, and steering controller.
You move the steering controller making sharp turns and stuff like you would normally. Its the old way of doing movement, but that doesn’t look realistic.
But then, you put the actual dog on a second object, that you would move and make follow the steering object, but with some inertia. Like not parenting the dog object to the steering object, but making him try to follow using forces, because smooth turns are caused by a change of momentum, implying forces…
Now, when you make your armature, you make so your armature follows the dog object (the one following the steering object), but you parent your IK controllers to the steering object.
The result is that the “main body” attached to the dog object will make smooth turns because of its inertia, but the limbs will follow the “desired trajectory”.
Its tricky, I get it. But I have no other idea lol