In our culture they don't allow dating a guy who is younger than a lady. They allow dating a guy who is older than a lady. So what should I do - should I move on and leave him or what ..?

There's no good reason why a woman shouldn't be able to date a man who is younger (especially if the age difference is small), and if this guy makes you happy then the relationship is probably worth a fight. Keep in mind also, that despite taboos, criticisms will likely fade over time if people see that you are happy and settled. However it's important to choose your battles wisely - if staying with him makes you more miserable than leaving him, then it's also okay to choose not to fight. Choose the path that leaves you with the least regrets.