applicationHost.xdt to specify that host header should be preserved and allow specific server variables (otherwise you get an exception). File is placed n the directory above wwwroot.

<configuration xmlns:xdt="http://schemas.microsoft.com/XML-Document-Transform">
    <system.webServer>
        <proxy xdt:Transform="InsertIfMissing" enabled="true" preserveHostHeader="true" reverseRewriteHostInResponseHeaders="false"/>
        <rewrite xdt:Transform="InsertIfMissing">
            <allowedServerVariables xdt:Transform="InsertIfMissing">
                <add name="HTTP_X_ORIGINAL_HOST" xdt:Transform="InsertIfMissing" xdt:Locator="Match(name)"/>
                <add name="HTTP_X_UNPROXIED_URL" xdt:Transform="InsertIfMissing" xdt:Locator="Match(name)"/>
                <add name="HTTP_X_ORIGINAL_ACCEPT_ENCODING" xdt:Transform="InsertIfMissing" xdt:Locator="Match(name)"/>
                <add name="HTTP_ACCEPT_ENCODING" xdt:Transform="InsertIfMissing" xdt:Locator="Match(name)"/>
                <add name="HTTP_HOST" xdt:Transform="InsertIfMissing" xdt:Locator="Match(name)" />
            </allowedServerVariables>
        </rewrite>
    </system.webServer>
</configuration>

web.config to specify how requests should be routed:

<system.webServer>
  <rewrite>
	<rules>
	  <rule name="Blog Proxy" stopProcessing="false">
		<match url="^blog(?:$|/)(.*)" />
		<action type="Rewrite" url="https://blog.mywebsite.com/{R:1}" appendQueryString="true" logRewrittenUrl="false" />
		<serverVariables>
		  <set name="HTTP_X_UNPROXIED_URL" value="https://blog.mywebsite.com/{R:1}" />
		  <set name="HTTP_X_ORIGINAL_ACCEPT_ENCODING" value="{HTTP_ACCEPT_ENCODING}" />
		  <set name="HTTP_X_ORIGINAL_HOST" value="{HTTP_HOST}" />
		  <set name="HTTP_ACCEPT_ENCODING" value="" />
		</serverVariables>
	  </rule>
	</rules>
  </rewrite>
</system.webServer>