Suppose that there is a hovercraft floating a few centimetres above the Earth's surface. As it is disconnected from the Earth, which is spinning from West to East, shouldn't it appear to move East to West to observers on the ground? Does this happen? If not, why not?
Let's make it even more interesting! If what you said were true, then all you'd need to do to travel large distances is jump up in the air. The Earth turns at a speed of around 500 m/s (at the equator) and so if you were up in the air for a second, you'd travel 500 metres! Well, obviously, you don't. (If you did, long-jump events would be much more exciting.)
The same thing could be said about trains like the TGV that travel at 100 m/s. Imagine you were sitting on one of these trains, and you dropped a coin and it took half a second to fall. It would hit the ground right below where you released it, not 50 m behind you. According to someone on the platform, the train, the coin, and you are all moving at the same horizontal velocity. When you release the coin, the force of gravity accelerates it vertically, but its horizontal velocity is unchanged as there are no forces in that direction.
You are currently travelling with the same angular speed as the Earth, and you continue to travel at that speed even when you're not "touching" the Earth's surface. When you jump, you are only changing your "vertical" velocity, not your "horizontal" velocity, which stays the same.
Compare the case of a plumb line. Anywhere on the planet a plumb line will hang perpendicular to local level surface. This is true both on a planet that is rotating, such as our Earth, and on a planet that is not rotating.
The Earth rotates at a constant angular velocity. For any object that is co-rotating with the Earth the speed is constant. Maintaining this constant speed does not require a force, only acceleration requires a force. Therefore a plumb line hangs perpendicular to the local level surface.