..
Pretty good example, but there are a few things you
can do with the code to make it even easier to help you,
see the altered source for further comments.  See also [1].

The problem is alluded to in my comments, see if
they make something 'click' for you.

<sscce>
// adding generic imports for classes assists
// people to see your GUI quickly
import java.awt.*;
import javax.swing.*;

public class WidgetPanel extends JPanel{

// Why are you keeping this reference to a JPanel?
// WidgetPanel *is* a JPanel!
// JPanel widget = null;

JButton back = null;
JButton forward = null;
JButton stop = null;
JTextField url = null;
JButton go  = null;

private static final long serialVersionUID = 12342113;

public WidgetPanel(){

//  widget = new JPanel(new GridLayout(2, 10));
 // just call 'super'
 super(new GridLayout(2, 10));
 setBackground(Color.blue);
 this.installFields();
}

private void installFields(){

 back = new JButton("<- BACK");
 // now add the components directly to the panel.
 add(back);

 forward = new JButton("FORWARD ->");
 add(forward);

 stop = new JButton("STOP");
 add(stop);

 url = new JTextField();
 add(url);

 go = new JButton("GO!");
 add(go);
}

// by adding the 'main' to WidgetPanel, the class and it's
// test are one and the same - bery portable.
public static void main(String[] args) {

 JFrame.setDefaultLookAndFeelDecorated(true);

 JFrame temp = new JFrame();
 WidgetPanel mypanel = new WidgetPanel();

 temp.setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

 temp.getContentPane().add(mypanel, BorderLayout.CENTER);

 temp.pack();
 // most well behaved GUI's do not need to be
 // sized, it is better to let the GUI find its
 // own size
 //temp.setSize(new Dimension(640, 100));
 temp.setVisible(true);
}
}
</sscce>

[1] Being such a fan of the SSCCE, I wrote this
document on them for further tips.
<http://www.physci.org/codes/sscce.jsp>

HTH

Funny, just yesterday someone came with some code which had almost the
same architectural problem. Have a look at the thread "Subpnel problem"
[sic].

Andrew already pointed out the main issues, I just have a few things:

Why do you do your own bar? Why not just use a JToolbar? They are made
for this.

Do you really plan to serialize this? Serialization of GUI (Swing)
components is usually not worth the effort and anyhow not guaranteed to
be compatible between Java releases.

See Andrew's remarks.

You might want to consider using icons for the arrows, instead of the
"ASCII art". You can mix an icon and text on a button, so that should
work nicely. If you need icons, the FAQ points to a number of icon
collections.

You might also want to consider to not use all caps (uppercase). No need
to shout at your users.

Sun has changed the recommended way to start a GUI. It is no longer
advisable to do it from the main thread. Instead you should do it from
the EDT. See Q4.3 of the FAQ for details.

pack() and then setting a fixed size don't work to well together. With
pack you tell layout management to calculate all the sizes and
positions. The setSize() works against this.

/Thomas