There could be many reasons why this is happening, but the first thing to do is to check how the training and validation dataset are built.
Is the dataset balanced? You said there are 5000 saples for 5 classes, but are they balanced? If, for example, the first class cover the 80% of your samples you may have this kind of overfitting behaviour.
Are tre training and validation sets equally split? If each class rougly represent a 1/5 of your samples (i.e. the classes are balanced), is this proportion preserverd in the training and validation set? It is not mandatory to have the exact percentages, but if in your training set class 5 only appear for the 1% of the samples and in the validation set you have a 20%, this again could explain your overfitting.
If the dataset is fine, you could try also to regularize the inputs before passing them into your network. If this is still is not enough, you can then focus on the network.