Get rid of the data-toggle="collapse" data-target="#navbar" attribute from
<div data-toggle="collapse" data-target="#navbar" id="navbar" class="navbar-collapse collapse">
Revised code:
<nav class="navbar navbar-default navbar-fixed-top">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#navbar" aria-expanded="false" aria-controls="navbar">
<span class="sr-only">Toggle navigation</span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="#">Project name</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li class="active"><a href="#">Home</a></li>
<li><a href="#about">About</a></li>
<li><a href="#contact">Contact</a></li>
<li class="dropdown">
<a href="#" class="dropdown-toggle" data-toggle="dropdown" role="button" aria-haspopup="true" aria-expanded="false">Dropdown <span class="caret"></span></a>
<ul class="dropdown-menu">
<li><a href="#">Action</a></li>
<li><a href="#">Another action</a></li>
<li><a href="#">Something else here</a></li>
<li role="separator" class="divider"></li>
<li class="dropdown-header">Nav header</li>
<li><a href="#">Separated link</a></li>
<li><a href="#">One more separated link</a></li>
</ul>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
<li><a href="../navbar/">Default</a></li>
<li><a href="../navbar-static-top/">Static top</a></li>
<li class="active"><a href="./">Fixed top <span class="sr-only">(current)</span></a></li>
</ul>
</div><!--/.nav-collapse -->
</div>
</nav>