A switch is only going to load balance the network activity like LACP. It's not going to load balance applications like you want.
The BEST option would be something like an F5 or Citrix Netscaler, but that's additional hardware. I know IIS has a software load balancing option, I'm sure something can be done to software load balance on linux as well.
Do you have the DB on the same server as the webserver? If so you may first want to consider offloading the DB to a separate server, that may free up enough resources for your web server to keep up with the demand.