You could use Custom Renderer to add the swipe event on specific platform. And handle them in Forms .
in Forms
create a CustomWebView
public class CustomWebView : WebView
{
    public event EventHandler SwipeLeft;
    public event EventHandler SwipeRight;
    public void OnSwipeLeft() =>
        SwipeLeft?.Invoke(this, null);
    public void OnSwipeRight() =>
        SwipeRight?.Invoke(this, null);
}
in Android
using Android.Content;
using Android.Views;
using App11;
using App11.Droid;
using Xamarin.Forms;
using Xamarin.Forms.Platform.Android;
[assembly: ExportRenderer(typeof(CustomWebView), typeof(MyWebViewRenderer))]
namespace App11.Droid
{
    public class MyWebViewRenderer : WebViewRenderer
    {
        public MyWebViewRenderer(Context context) : base(context)
        {
        }
        protected override void OnElementChanged(ElementChangedEventArgs<WebView> e)
        {
            base.OnElementChanged(e);
            Control.SetOnTouchListener(new MyOnTouchListener((CustomWebView)Element));
        }
    }
    public class MyOnTouchListener : Java.Lang.Object, Android.Views.View.IOnTouchListener
    {
        float oldX;
        float newX;
        CustomWebView myWebView;
        public MyOnTouchListener(CustomWebView webView)
        {
            myWebView = webView;
        }
        public bool OnTouch(Android.Views.View v, MotionEvent e)
        {
            if (e.Action == MotionEventActions.Down)
            {
                oldX = e.GetX(0);
            }
            if (e.Action == MotionEventActions.Up)
            {
                newX = e.GetX();
                if (newX - oldX > 0)
                {
                    myWebView.OnSwipeRight();
                }
                else
                {
                    myWebView.OnSwipeLeft();
                }
            }
            return false;
        }
    }
}
in iOS
using App11;
using App11.iOS;
using Foundation;
using UIKit;
using Xamarin.Forms;
using Xamarin.Forms.Platform.iOS;
using ObjCRuntime;
[assembly: ExportRenderer(typeof(CustomWebView), typeof(MyWebViewRenderer))]
namespace App11.iOS
{
    public class MyWebViewRenderer:WkWebViewRenderer
    {
        protected override void OnElementChanged(VisualElementChangedEventArgs e)
        {
            base.OnElementChanged(e);
            if(e.NewElement!=null)
            {
                this.BackgroundColor = UIColor.Red;
                UISwipeGestureRecognizer leftgestureRecognizer = new UISwipeGestureRecognizer(this,new Selector("SwipeEvent:"));
                leftgestureRecognizer.Direction = UISwipeGestureRecognizerDirection.Left;
                UISwipeGestureRecognizer rightgestureRecognizer = new UISwipeGestureRecognizer(this, new Selector("SwipeEvent:"));
                rightgestureRecognizer.Direction = UISwipeGestureRecognizerDirection.Right;
                leftgestureRecognizer.Delegate = new MyWebViewDelegate();
                rightgestureRecognizer.Delegate = new MyWebViewDelegate();
                this.AddGestureRecognizer(leftgestureRecognizer);
                this.AddGestureRecognizer(rightgestureRecognizer);
            }
        }
        [Export("SwipeEvent:")]
        void SwipeEvent(UISwipeGestureRecognizer recognizer)
        {
            var webview = Element as CustomWebView;
            if(recognizer.Direction == UISwipeGestureRecognizerDirection.Left)
            {
                webview.OnSwipeLeft();
            }
            else if(recognizer.Direction == UISwipeGestureRecognizerDirection.Right)
            {
                webview.OnSwipeRight();
            }
        }
    }
    public class MyWebViewDelegate: UIGestureRecognizerDelegate
    {
        public override bool ShouldRecognizeSimultaneously(UIGestureRecognizer gestureRecognizer, UIGestureRecognizer otherGestureRecognizer)
        {
            return false;
        }
    }
}
Now you just need to use it like 
<local:CustomWebView x:Name="browser"
           HorizontalOptions="Center"
           VerticalOptions="CenterAndExpand" 
           SwipeLeft="browser_SwipeLeft"
           SwipeRight="browser_SwipeRight">