I don't want to steal any credit from another answer, but I wanted to elaborate on my comment. I re-installed Visual Studio (long-shot I know) and still had the issue. It seems that when the HTTP modules for AI are loaded into IIS Express things go south quickly, so I had to resort to only loading those modules when running the release configuration.
This means updating your web.config to remove the AI statements, and instead move them to Web.Release.config as transforms so they're loaded when a release configuration is built:
https://stackoverflow.com/a/27923364/571237
Note however that the assemblies have changed since that answer was posted. Here's what I needed to add:
<system.web>
<compilation xdt:Transform="RemoveAttributes(debug)" />
<httpModules>
<!-- Enable application insights when running in release mode (it don't work right locally...) -->
<!-- https://stackoverflow.com/a/27923364/571237 -->
<add name="ApplicationInsightsWebTracking" type="Microsoft.ApplicationInsights.Web.ApplicationInsightsHttpModule, Microsoft.AI.Web"
xdt:Transform="Insert"/>
</httpModules>
</system.web>
<system.webServer>
<modules>
<!-- Enable application insights when running in release mode (it don't work right locally...) -->
<!-- https://stackoverflow.com/a/27923364/571237 -->
<remove name="ApplicationInsightsWebTracking" xdt:Transform="Insert"/>
<add name="ApplicationInsightsWebTracking" type="Microsoft.ApplicationInsights.Web.ApplicationInsightsHttpModule, Microsoft.AI.Web" preCondition="managedHandler" xdt:Transform="Insert" />
</modules>
</system.webServer>