This solved my problem:

let config = WKWebViewConfiguration()
config.userContentController = contentController
config.preferences.setValue(true, forKey: "allowFileAccessFromFileURLs")
webView = WKWebView(frame: .zero, configuration: config)
webView.uiDelegate = self
webView.navigationDelegate = self
view = webView

For those who came though SO to find this topic:

This isn't official to turn off the security in WKWebView, but we could use the private API to allow this just like this guy did for the Cordova project: cordova-plugin-wkwebviewxhrfix

The clue is to create the configuration for the WebView and set the allowFileAccessFromFileURLs property.

WKWebViewConfiguration* configuration = originalImpSend(_self, selector, settings);

// allow access to file api
@try {
    [configuration.preferences setValue:@TRUE forKey:@"allowFileAccessFromFileURLs"];
}
@catch (NSException *exception) {}

@try {
    [configuration setValue:@TRUE forKey:@"allowUniversalAccessFromFileURLs"];
}
@catch (NSException *exception) {}

return configuration;

But as I mentioned this is the private API and it could be a reason to reject your application in Apple's App Review. If you want to publish your app in App Store, please consider to run some lightweight HTTP server instead of violate overall security of the web view. Example: GCDWebServer.

So the accepted was not working for me when I first tried it (I was doing it wrong at the time) and the bug about (https://bugs.webkit.org/show_bug.cgi?id=154916) it strongly recommends against doing it. My solution was to implement a custom url scheme and use that to load all files. Looks like this.

First creating the WKWebView to which the url scheme is attached (you must create the WKWebView yourself, it can't be created on a storyboard).

override func viewDidLoad() {

    let configuration = WKWebViewConfiguration()
    configuration.setURLSchemeHandler(PrayerAssetHandler(), forURLScheme: "x-file")

    webview = WKWebView(frame: webContentView.bounds, configuration: configuration)
    self.webContentView.addSubview(webview!)
    webview?.autoresizingMask = webContentView.autoresizingMask
    webview!.navigationDelegate = self

    super.viewDidLoad()
}

Then implement the handler.

import Foundation
import WebKit
import MobileCoreServices

class PrayerAssetHandler: NSObject, WKURLSchemeHandler {

    func webView(_ webView: WKWebView, start urlSchemeTask: WKURLSchemeTask) {
        let url = urlSchemeTask.request.url!

        let pathArr = url.path.components(separatedBy: ".")
        let forResource: String = pathArr[0]
        let ofType: String? = pathArr.count > 1 ? pathArr[1] : nil

        let bundlePath = Bundle.main.path(forResource: "data_sub_folder" + forResource, ofType: ofType)

        let fileExtension: CFString = ofType! as CFString
        guard
            let extUTI = UTTypeCreatePreferredIdentifierForTag(kUTTagClassFilenameExtension,
                                fileExtension, nil)?.takeUnretainedValue()
        else { return }

        guard
            let mimeUTI: NSString = UTTypeCopyPreferredTagWithClass(extUTI,
                                kUTTagClassMIMEType)?.takeRetainedValue()
        else { return }

        let mimeType: String = mimeUTI as String

        do {
            let data: Data = try NSData(contentsOfFile: bundlePath!) as Data

            //Create a NSURLResponse with the correct mimetype.
            let urlResponse = URLResponse(url: url, mimeType: mimeType,
                                          expectedContentLength: data.count, textEncodingName: "utf8")
            urlSchemeTask.didReceive(urlResponse)
            urlSchemeTask.didReceive(data)
            urlSchemeTask.didFinish()
        }  catch _ as NSError {
            return
        }

    }

    func webView(_ webView: WKWebView, stop urlSchemeTask: WKURLSchemeTask) {
    }
}