screen window exits when the program it is running exits. This is because screen is itself not a shell, so once the program exits, there is nothing more that can be done with the window. You cannot launch a new program, as there is nothing remaining under that terminal session that could launch a program.
A newly created screen window will by default run bash (or whatever your nominated shell is). As bash is a shell, you can launch any number of programs from it, and when they exit, they return to bash. Then another program can be launched. So a window running bash will behave as you describe. It is only when bash itself exits will the window close.
Note that this is the same behaviour as any terminal session, once the last program exits, the terminal session ends (with a normal terminal, this session is then respawned if set to by init and you are then presented with a login screen).
This is a bit hacky, but normally when you start screen with a command, the command completing ends the window. However, if you also start bash with command using the
-c option, bash will also exit once the command exits.
So if we want bash to run a command but stick around once the command ends, we need to run the command as part of the bash initilisation process.
For example, if you create a new file "startserver1" containing the commands needed to start the server:
#start the server
Then run screen with bash, initialising with the startserver1 file:
screen bash --init-file startserver1
This will invoke screen, starting bash, initalising bash with your server start script. Once the server ends, you are left in the bash session within the screen window.