wp installation in subfolder of one server now direct new url to it

Hello and sorry if this question has been asked here before. I couldn´t find any answere to my problem but I have a horrible conection here in the Solomons so please excuse.

I have a bluehost server with a domain and run a wordpress installation on it. http://www.mywebsiteaaa.com Now I want to have another website for my club. so I installed wordpress in a subfolder on that server again and set up a new website. the url would be http://www.mywebsiteaaa.com/clubswebsite

so far so good. but now I bought a new domain for the clubs website. http://www.clubsdomain.com
The is no webspace with it so I can only do a redirect.

now if someone visits types in http://www.clubsdomain.com he will see the website correctly but here is my PROBLEM.
the url doesnt change when you click on different pages or posts on that website for example /projects/project1 Whatever you click on in the browser it will always just say http://www.clubsdomain.com. so ifsomeone would hit refresh he would not refresh the page he is on he will always be back on home …

Any ideas ? What I want is that a page that would normaly be http://www.mywebsiteaaa.com/clubswebsite/projects/project1 would be http://www.clubspage.com/projects/project1

an NOT just http://www.clubspage.com

Please help me out with my most likely for you guys “little” problem … thanks in advanced and greetings from the solomons.
cheers