Swift Segmented control Switch multiple views from navigation Title

At first, if you're not using Storyboards/XIBs/InterfaceBuilder and doing everything programmatically, you can remove @IBAction, as it means Interface Builder Action.

Secondly, it is good idea to put segmented control as a variable, so you can access it easier when needed:
var customSC: UISegmentedControl - just after class MyVC: UIViewController {.
But Swift requires all variables to be initialized before calling a superclass designated initializer - init(nibName nibNameOrNil:, so, let's initialize it: it is possible to initialize variables in it's declaration:

Change var customSC: UISegmentedControl to

  var customSC: UISegmentedControl = {
    let items = ["Tab 1", "Tab 2", "Tab 3", "Tab 4"]
    let control = UISegmentedControl(items: items)
    control.selectedSegmentIndex = 0
    return control
    }()

So, we don't need to configure Segmented Control in viewDidLoad method.
The problem, why you your control wasn't working is that you haven't added target to it. So, when you change selected segment, the control does not sends any events to your viewcontroller. Let's fix it. Add this method to the end of viewDidLoad:
customSC.addTarget(self, action: "indexChanged:", forControlEvents: UIControlEvents.ValueChanged)
Now, when you press any of the segments, the method indexChanged(sender: UISegmentedControl) will be called and the view controller will be able to react.
It would be easier to put segmented control on top of navigation bar this way: navigationItem.titleView = customSC, it is much shorter than this: navigationController?.navigationBar.topItem?.titleView = customSC

When using Interface Builder it is OK to use weak modifiers for WebView as it is being automatically added to the main view of the controller and being held in memory. But if you add it programmatically, you should avoid using weak in this context in order to prevent undesired behavior (Web View will be deallocated as soon as there are no strong (i.e. not weak) links to it, so basically, it will be deallocated immediately after creation and you end up with BAD ACCESS error. To fix it, just change this: weak var webView: UIWebView! to this var webView: UIWebView!.

So, after small refactoring, we may finish with something close to this:

import UIKit
class MyVC: UIViewController {

  var webviewURL: NSURL = NSURL(string: "myURL")!
  var webviewURL2: NSURL = NSURL(string: "myURL")!
  var webviewURL3: NSURL = NSURL(string: "myURL")!
  var webviewURL4: NSURL = NSURL(string: "myURL")!

  var webView: UIWebView!

  var customSC: UISegmentedControl = {
    let items = ["Tab 1", "Tab 2", "Tab 3", "Tab 4"]
    let control = UISegmentedControl(items: items)
    control.selectedSegmentIndex = 0
    return control
    }()

   func indexChanged(sender: UISegmentedControl) {
    switch customSC.selectedSegmentIndex { //error here
    case 0:
      let requestObj = NSURLRequest(URL: webviewURL)
      webView.loadRequest(requestObj)
    case 1:
      let requestObj = NSURLRequest(URL: webviewURL2)
      webView.loadRequest(requestObj)
    case 2:
      let requestObj = NSURLRequest(URL: webviewURL3)
      webView.loadRequest(requestObj)
    case 3:
      let requestObj = NSURLRequest(URL: webviewURL4)
      webView.loadRequest(requestObj)
    default:
      break;
    }
  }

  override init(nibName nibNameOrNil: String?, bundle nibBundleOrNil: NSBundle?) {
    super.init(nibName: "myVC", bundle: nil)
  }

  convenience init() {
    self.init(nibName: "myVC", bundle: nil)
    //set the tab bar item's title
    tabBarItem.title = "title"

    //put an image on the tab bar item
    tabBarItem.image = UIImage(named: "image")
  }
  required init(coder aDecoder: NSCoder) {
    fatalError("init(coder:) has not been implemented")
  }


  override func viewDidLoad() {
    super.viewDidLoad()

    let logo = UIImage(named: "image")
    let logoView = UIImageView(frame: CGRect(x: 0, y: 0, width: 100, height: 80))
    logoView.contentMode = .ScaleAspectFit
    logoView.image = logo
    let leftLogo = UIBarButtonItem(customView: logoView)
    self.navigationItem.leftBarButtonItem = leftLogo

    navigationItem.titleView = customSC
    navigationController?.navigationBar.translucent = false

    let requestObj = NSURLRequest(URL: webviewURL)
    webView.loadRequest(requestObj)
  }
}