What you describe is not possible. But there’s still good news:
What is possible however is to establis a Dynamic connection with the SSH Server. This will open a port on your local computer to which you can point the Proxy setting of your Browser and allow you to use the tunnel as a proxy server. But you have to type a hostname/ip and port into the browser as if the browser were running on the machine the SSH Server is on.
Command looks like this: ssh user@server.example.com -D 1234
Then point your browser’s proxy to localhost:1234.
So if you tunnel into Server A, and want to connect to server B, you type into your browser whatever address you would type into a Browser running on Server A. If a browser running on server A could not connect to Server B (if the process on Server B only listens on 127.0.0.1) then you still couldn’t connect. It sounds like you just have the one server, but I wanted to be sure this was clear.
If you just have the one server, you tunnel into it with the Dynamic connection, set your proxy. You will then be able to type “localhost:1234” (for example) into the browser and it will connect to the service running on the remote server on port 1234.
Securit Side Note: Never never never setup a server where root can SSH in! Serious security flaw. Create a normal user account (who is allow to su or sudo) and SSH in as that user.