Domain Forwarding

Say I’ve set up a wordpress website at https://www.example.com/~user/wordpress and also purchased the domain coolexamplesite.com, and I’ve domain forwarded it to the above wordpress directory. Everything is working perfectly – https://www.coolexamplesite.com takes you to the website and there are no problems, except that the links show up as the original example.com address. Is there anyway within wordpress to have it only show the forwarded domain?

Thanks greatly in advance!