Does your website have a web.config file? It should if your website is being hosted on Windows.
If so the reason your website is probably crashing is because the web.config files in both of your sites have the same name for your rewrite rule that is used for the permalinks.
I was having this issue too and to fix it I went into my subdirectory site and replaced the contents of the web.config file with the following:
<?xml version="1.0" encoding="UTF-8"?>
<configuration>
<system.webServer>
<rewrite>
<rules>
<clear />
<rule name="Some different name" stopProcessing="true">
<match url=".*" />
<conditions logicalGrouping="MatchAll">
<add input="{REQUEST_FILENAME}" matchType="IsFile" negate="true" />
<add input="{REQUEST_FILENAME}" matchType="IsDirectory" negate="true" />
</conditions>
<action type="Rewrite" url="index.php" />
</rule>
</rules>
</rewrite>
</system.webServer>
</configuration>
I then reuploaded it to the subdirectory site and that fixed it.
Note: A web.config file is only created when using permalinks. It is not created when using the default links like p?=123 so I just selected a Permalink option in the admin area just to create the web.config which I then did the above.
I hope this helps.