What could be possible causes and solution starting from small to big..
Thank you in Advance,
There are various factors that can contribute to steering while vibration at higher speeds. Sometimes vibrations are due to minor reasons such as a wheel balance weight falling off causing wheel balance issues, but it is not always the case. For example, a used car with tons of miles where the suspension components are questionable is highly likely to experience a similar issue. You have not mentioned whether your car is new/used, but I am just gonna talk about all the factors that can cause this issue.
Before you throw money to a mechanic, there are few basic checks which you can do your own. Check all four wheels and make sure that the lug nuts/bolts are still tight. If this is good, then check all four tire surfaces for uneven tread wear. If any of the tires are cupped(if you are not sure what tire cupping is, do a google search for images) it can also cause lead to vibrations. If you see any tire cupping, chances are high that the wheel alignment is off on your car in which case you need to replace the cupped tire and do an alignment afterwards.
The other two checks need you to jack up the car and then check each wheel individually for excessive play. You can skip these if you are not mechanically inclined.:) Excessive wheel play in the front wheels are caused by two factors; bad wheel bearings or bad inner tie rod ends. Where as in rear wheels, bad wheel bearing is the only cause for excessive play. To check for a bad tie rod end on front wheels, grab the wheel at 3 and 9 O'clock position with your hands and then move it in and out with force while it is on air. If you feel movement, then the tie rod ends are bad and is the reason why steering wheel is shaking at higher speeds. To check for a bad wheel bearing, grab the wheel at 6 and 12 O'clock position and then move it in and out with force. If you see movement, then the wheel bearing is bad and is the cause of vibration. Do the same wheel bearing check for the rear wheels as well. Driving on a failed wheel bearing is dangerous, it will cause uneven tire wear/vibrations and can kick out wheels from the hub while driving if it is in real bad shape.
If you did not find anything yet, then give the car to a tire shop to check the wheel balancing. I am pretty sure that the tire shops in India will also check for any bend in the wheels as well. A slight bend in any of the wheels can also cause similar problems. They might also do a tire rotation afterwards.
After all these service from the tire shop if you still experience vibrations, then your car has got a problem with the suspension such as worn struts or strut bearing/ mounts etc. Usually the tire shops will be able to check these as well, but am not sure how the tire shops there operates. You can ask them whether they will inspect suspension components as well, if not give it to a regular car mechanic.