Does the energy flow during interference violate the existence of speed limit? I was recently told by my physics teacher that when two waves interfere then the energy lost in the destructive region is transferred to the constructive region and hence no violation of conservation of energy.
Now the speed of EM waves is the speed limit of the universe. Keeping this in mind , we can conclude that when two EM waves interfere , for energy to be transferred from destructive region to the constructive region, the energy must go faster than its initial traveling speed (the speed of light) so as to cover up the extra distance but this can't be true since not even light can cross its speed (right ?).
Where am I making a mistake ? How does the energy actually gets transferred and at what speed (for EM waves) ?
 A: 
for energy to be transferred from destructive region to the constructive region, the energy must go faster than its initial traveling speed (the speed of light) so as to cover up the extra distance

This is incorrect. The key problem is the idea of “extra distance”. There is no extra distance. A region of interference only exists where EM waves from both sources have already arrived. This is never ahead of either of the two sources wavefronts and is often well behind one of them. There is no risk at any time of superluminal energy transfer to regions of destructive interference because all regions of interference (both constructive and destructive) are always within the past light cone of both sources.
A: In classical electromagnetic waves that are described by solutions of the Maxwell equations, the energy carried by the wave is given by the Poynting vector, as a rate of energy transport

and is in term of averages per unit area, and cannot be used for determining any direction in energy,afaik.
To answer your question, one has to go to the quantization frame where  a light beam is composed out of zillions of photons. Photons also go with velocity c.
The simplest demonstration of interference patterns and how single photons build up the double slit interference experiment may help:


Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames

Note that the little dot footprint each photon  leaves is at an angle with the center of the slits, and yes, the photon travels a  larger distance than the straight line of a classical light ray. Photons still travel at speed c, so individual photons take longer to hit the screen, BUT, photons are not classical light, in a similar way that bricks build bridge  but the bridge  is not a brick. They build up quantum mechanically the classical electromagnetic radiation in a complicated quantum field theory , QED, way.
This is discussed  in this link How classical fields, particles emerge from quantum theory is discussed.
Interference of classical light and quantum phenomena have  unexpected  consequences as can be seen in this MIT video.
A: I was recently told by my physics teacher that when two waves interfere then the energy lost in the destructive region is transferred to the constructive region and hence no violation of conservation of energy.
This is exactly how constructive and destructive interference work together. For mechanical waves. In areas of destructive interference, the up and down movement of particles in opposite directions causes an increase in density and the particles are pushed away to the side in the direction of the reduced density in constructive interference.
What does the picture look like with electromagnetic radiation?
...we can conclude that when two EM waves interfere, ...energy to be transferred from destructive region to the constructive region,...
For EM radiation, too, there is a density, the radiation density. However, a displacement of photons from areas of higher density into such lower density does not take place. Simply because in the energy range, in which these experiments take place, an interaction between photons is a more than rare event.
The concept of interference is too much of a simplification for EM radiation to hold up in the realm of physics. And I also think it is not correct to present this in school lessons. (But of course students have the duty to reproduce correctly the interpretation taught in this way. This is how the inquiry of a learning process works).
The intensity distribution of the EM radiation behind edges is the result of the deflection of this radiation at these very edges. Here an interaction takes place. Without edge no intensity distribution on an observation screen.
A: In addition to the other answers, I'd like to point out something important:
Your beam of light with an interference pattern does not travel at the speed of light!
The speed of light $c$ is not the speed of a beam of light of any size and profile, it is the speed that an infinite plane wave of light that's existed throughout time would move.
As discussed in @flippiefanus's answer to Wouldn't any structured beam of light be expected to travel slower than a plane wave? once you have transverse modulation like an interference pattern and/or standing waves, the longitudinal propagation is slower than the speed of light.
For further details I refer the reader to the linked answer.
A: In the DSE there are no photons (i.e. NO energy) in the dark areas. All the photons (i.e. all the energy) goes to the bright areas. In high school you are taught the simplest theory .... but if you think about it you can see the flaws. Richard Feynman and Dirac stated every photon determines its own path, Feynman discovered the "path integral theory" to explain why light chooses a path.
