How electron movement produces current,instead of having a slow drift speed?
An analogy

Imagine pressing 1 cm ball-bearings into a 10 m long horizontal pipe (1.1cm internal diameter) which is full of identical ball-bearings.

You take 1s to slowly press one new bearing into your end of the pipe

That 1 cm ball bearing travelled at 0.01 m/s

As you press a new ball-bearing in at your end, another ball bearing gets pushed out the far end 10 m away.

That effect travelled at nearly 10m/s

The pressure wave that caused that effect travelled even faster (probably at near the speed of sound in ball-bearing material less some allowance for small gaps between bearings, lateral movements and other messy and irrelevant stuff).

The trouble with analogies

This analogy may help you to think about the different things that are happening at different scales and that they are not related in quite the way you may have thought. Analogies can only go a short way to explaining these subjects.

A Tsunami wave in the ocean, travels at around 800 Km/h (500 MPH) - but the water does't flow that fast.

An electrical signal can travel much faster than electron drift velocity.

Electricity

Electricity is not (simply) a flow of charge carriers. It also involves an electric field. Changes in that field can propagate much faster than charge carriers can move.

Follow-up


  throughout the wire, there exist a big number of electron


Throughout all matter, there exist a large number of electrons. This includes air and other materials normally considered insulators - through which there is normally no appreciable current. In metals, the presence of so-called "free electrons" is important, but these are usually only a small proportion of the electrons in the metal.


  negative terminal of the battery is having electrons which are having high energy


No, for the purposes of this discussion, they have nearly the same energy as all the other electrons in the wire. The electrons do not carry energy from the battery.

This isn't a useful way to think about electricity. If you wanted to understand the Physics of electricity, Physics.stackexchange would be a good place to do some searches.



See also 


How fast does electricity flow? 
Current flow in batteries
Does an electron travel thousands of miles