One of the more common problems that drivers may notice in their vehicle is a shaky steering wheel, especially at high speeds. This issue can be a bit nerve-wracking the first time a driver feels it, but it's not all that rare. That being said, it is something that you should look into as soon as possible.
There are many different issues that could cause a steering wheel to shake at high speeds, but the most common is likely a bent wheel. This doesn't refer to the steering wheel itself, but the wheels underneath the car that the tires are put on. Unfortunately, you pretty much need to get your car off of the ground in order to check this. You can have a mechanic lift your car up or use a jack yourself if you're comfortable doing so. Spin each wheel and look for any sideways movement, as this typically indicates a bent wheel.
Wheel bearings are thought of as life-of-the-car parts, but it's not unheard of for them to become damaged, which very well could lead to a vibrating steering wheel. While you have the car up off the ground, see if you can push the wheels in and out. If the wheels start to move back and forth, that likely indicates that the bearing needs replacement.
If all your wheels are perfectly straight and the bearings are up to par, then it doesn't hurt to get them balanced or rebalanced. This is typically enough to cure most problems related to shaky steering wheels, but if the issue lies elsewhere, it won't do much. Fortunately, rebalancing your wheels every now and again is a good idea anyways, so it doesn't hurt to try that early.
At this point, it makes sense to look at other parts of the vehicle. The suspension also plays a big role in steering, and there are many shocks and struts designed to minimize the impact of the road. But since you're feeling the impact of the road, it stands to reason the suspension is not doing its job properly, assuming the wheels are fine. Steering dampers are the first area that drivers should look into when it comes to feeling vibrations, but it could also be tie rods or any number of other worn or loose suspension components.
If your steering wheel shakes not while speeding up, but while braking, then the problem likely isn't related to your wheels or your suspension. Instead, the most probable culprit is a warped brake rotor. These critical parts get worn out of shape and provide feedback to the steering wheel as the car slows to a halt.