routing internal users to web servers in DMZ

My internal users are on private IPs (192.168.50.x/24), the web servers are on private as well(192.168.10.x/24) There is an ASA 5520 seperating them and the internet.

The issue is that when an internal user trys to hi our web server using the proper domain name it trys to get to the public IP which does not work because you cant use the same pipe out and in.

My question is that I need to know where to configure in the ASA to redirect these requests inside prior to them leaving the network or should this be corrected using pointers on the DNS server?

Thanks,