When you run the server, or any process, it is attached to the terminal. When the terminal dies, it dies too.
To keep the server running without you keeping the putty window open, you have to run it detached from said putty window. As we assume you want to keep access to the process so you can administrate the server, you need to use tools such as GNU screen.
Easiest way:
Log in as you.
type 'screen -S server1'
it should then pop up with the same prompt as before etc.
do 'exec su server1s username' to switch user.
cd to the right folder.
start the server as per normal, with exec in front of it. (so 'exec tremded +net_port 12345 +whatever)
do Ctrl+A+D and it should go back to your old prompt.
log out, log back in, do whatever else, then do 'screen -r server1' and it should be back to your server.
Just be careful to do ^AD when your finished watching the console, not ^C, killing servers is no fun.