Можете ли вы попробовать следующее,
...
<WebView
source={{
uri: "https://************.org/privacy-policy/"
}}
ref={WEBVIEW_REF => (this.WebViewRef = WEBVIEW_REF)}
javaScriptEnabled={true}
injectedJavaScript={
"document.getElementsByClassName('site-content')[0].style.paddingTop='0px',document.getElementsByClassName('header')[0].style.display='none', document.getElementsByClassName('footer')[0].style.display='none',document.getElementsByClassName('subscribe-line')[0].style.display='none'"
}
onLoadProgress={e => this._onLoading(e.nativeEvent.progress)}
/>
...