Its funny you mention this. I noticed this too close to when I bought the car. The build quality of this car has come into question many times, I hear NUMEROUS squeaks, creaks, rattles, and vibration noises that I actively keep track of to report to the dealer on first service. After the installation of this strut bar, I have to say that a lot of those squeaks and rattles that I used to hear from the front suspension have stopped. The car seems to return to its "equilibrium position", what I mean by this is the way it drives on a smooth road, much quicker then it used to. The steering wheel, well thats still there but its expected, let me explain this also: I have a similar car to your Tucson, a CR-V. These SUVs are geared to give a more comfortable ride hence they provide less info about the road in the steering wheel. The Sonata, especially the ones running 18s are expected to give feedback in the steering wheel about bumps on the road because of the lower profile tires and the stiffer springs and shocks. What I was referring too about "the steering going loose" was that at some speeds it would feel like the cars front end was actually HOPPING over bumps. In fact on fast left handers, like when making a left on a yellow, I could actually feel the car hopping and losing traction. It was actually scary. Luckily I believe that most of the problems that the Sonata has with the front suspension have been fixed with a strut bar, however one word of advice: hold off on getting the $220 Solid Strut from SR, after installation I am experiencing a really annoying and cheap sounding rubbing noise at low rpms. I believe this has to do with the strut bar rubbing against the head of the fire wall of the car. Get either of the other 2 strut bars from SR, they accomplish the same basic tasks but manage to stay away from the cars firewall, at least as far as I can tell from the pictures.
Hope that helped.