How do I make DrawerLayout to display below the Toolbar?

Change Your drawer layout style like as below

RelativeLayout
 ----Toolbar
 ----DrawerLayout
     ---ContentView
     ---DrawerList 

<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
android:orientation="vertical"
android:layout_width="match_parent"
android:layout_height="match_parent">

    <!-- The toolbar -->
    <android.support.v7.widget.Toolbar  
        android:id="@+id/my_awesome_toolbar"
        android:layout_height="wrap_content"
        android:layout_width="match_parent"
        android:minHeight="?attr/actionBarSize"
        android:background="?attr/colorPrimary" />

    <android.support.v4.widget.DrawerLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    android:id="@+id/my_drawer_layout"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

        <!-- drawer view -->
        <LinearLayout
            android:layout_width="304dp"
            android:layout_height="match_parent"
            android:layout_gravity="left|start">

            <!-- drawer content -->

        </LinearLayout>

        <!-- normal content view -->
        <LinearLayout
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            android:orientation="vertical">



            <!-- The rest of content view -->

        </LinearLayout>  
    </android.support.v4.widget.DrawerLayout>

</LinearLayout>  

i don't think you can when using custom toolbar

but a work around would be to set a top_margin to drawer. (the same thing on google play store!)

<!-- drawer view -->
<LinearLayout
    android:layout_marginTop="?attr/actionBarSize"
...

if you found a better solution tell me too ;)