Progress bar with rounded corners?

How to make the progress shape a bit smaller?

You need to give the progress item a little padding, like so:

<item android:id="@android:id/progress"
    android:top="2dp"
    android:bottom="2dp"
    android:left="2dp"
    android:right="2dp">

How to make the progress bar to be curved as per design?

Replace the <clip></clip> element, with <scale android:scaleWidth="100%"></scale>. That will make the shape keep its form as it grows – and not cut off.
Unfortunately, it will have a little unwanted visual effect at the beginning – as the shape corners don’t have enough space to draw. But it might be good enough for most cases.

Full code:

<layer-list xmlns:android="http://schemas.android.com/apk/res/android">
 <item android:id="@android:id/background">
    <shape>
        <corners android:radius="8dp"/>
        <solid android:color="@color/dirtyWhite"/>
    </shape>
 </item>

 <item android:id="@android:id/progress"
    android:top="1dp"
    android:bottom="1dp"
    android:left="1dp"
    android:right="1dp">

    <scale android:scaleWidth="100%">
        <shape>
            <corners android:radius="8dp"/>
            <solid android:color="@color/colorPrimaryDark"/>
        </shape>
    </scale>
 </item>
</layer-list>