String is a constructor that's builtin to JavaScript. Naming variables that shadow these constructors will cause an error:
TypeError: String is not a constructor
Rename the String variable and do not use the switch statement here because you have a String instance. switch statements use strict comparison (===) per the MDN documentation and the ECMAScript 2015 specification. Since a string instance and literal are never 'strictly equal', comparison fails. Don't instantiate, instead use a literal:
var string = "String #1";
switch(string) {
  case "String #1":
    document.write("String Number 1");
    break;
  default: 
    document.write("wrong string");
}
 
 
Also, I don't recommend using document.write, see here. Logging or inserting into the DOM with createElement and appendChild should work sufficiently here.