It sounds like you need a way to keep an eye on your site's performance from the end-user's perspective.
One way to do this is to have your client-side scripts include a way to log to a log aggregation site like SumoLogic. Here is a doc to reference about using client-side JavaScript to log to SumoLogic.
On your server side, you could implement a /ping
API endpoint that would just immediately return true
so you know how long it takes your user to at least reach your site. You can then log to SumoLogic how long that request took. You could do this with other requests as well to see which APIs are slower than others.
If you include geo-location when logging to SumoLogic, you can see how well your site performs around the world.
And if you want to get really fancy, then you should implement a custom header that your APIs understand which is a transaction token of some sort for all requests. When your server receives that header, it should use that token throughout the request's logs so you can see where things go wrong and what to do about them.
Another good site to check out for this sort of thing is New Relic - Browser Monitoring. This is much more performance-centric and you don't get the insights of your own logs, but it's an awesome app in its own right.
EDIT
As mentioned in the comments by @Bergi, you could also have your server respond with the headers immediately and measure performance that way.