You can have a look at the sample project from the link below. This is working as expected, please let me know what you are doing differently so I can further investigate this issue.
I have extracted the essential parts from my new application into a demo project. I hope I have removed all the non-essential references so it is easily compiled. There is a set of inherited forms starting with miw_BaseForm ending at wfmMenuLeft. The wfmBase.pas unit has an OnResize event which stops the menus working. If I remove it, all is OK.
Just hover the mouse over the top button at the left and click on the menu.
If you have a better option for implementing the buttons and menus at the left, I would be glad to hear it.
The example.html is a saved file from the browser when running the application.
From what I can see in your sample project you are trying to show/hide the sidemenus when clicking on an IWAdvImage. I'm getting several JavaScript errors that do not seem to be related to the TMS IW controls.
I would suggest using a IWPopupMenuLabel instead and define the image source in the Caption property.