What you're proposing isn't the use-case for Application Gateways. Application Gateways are Layer 7 load balancers / reverse proxies. What you want to do is almost treat them as a one-site forward proxy. It's not a good architecture and even if it were possible would ultimately be more costly in the long-run since you would pay for data egress as your App Gateway accepts requests and then forwards on to your web servers via an outbound connection over the Internet. They then receive the response headers/body from your web servers and again send that result on to the original caller.
In that scenario, you are forced to have to use end-to-end SSL for your applications, removing any possibility of using the App Gateway for SSL offload in the future. If your traffic isn't encrypted or doesn't need to be, the predictability of the source and destination of your traffic increases the security risk to your website's users and your company.
You also have the possible security implications of this type of architecture. Your web servers still need to be accessible at the very least by your Application Gateway, which means they are either freely available on the Internet anyway (in which case why bother with an App Gateways at all) or they're firewalled at a single layer and permit only traffic from the source IP address of your Application Gateway.
The bad news with the firewall approach is that you cannot assign a static public IP address to an Application Gateway, it is forced as Dynamic. Realistically the public IP won't change until the App Gateways are rebooted but you should know that when, not if, they do, your firewall rules will be wrong and your App Gateways won't be able to get to your DMZ servers any more, which means an outage. The only true solution for that is a firewall that can do URI based firewall rules...the impact there is cost (time and CPU) to perform a DNS lookup, see if the traffic is from the App Gateway by its DNS address - something like bd8f86bb-5d5a-4498-bc0c-e1a48b3873bf.cloudapp.net and then either permit or deny the request.
As discussed above, a further security consideration is that your traffic will be fairly consistently originating from one location (the App Gateways) and arriving at your DMZ. If there's a well defined source of traffic, that fact could be used in an attack against your servers/DMZ. While I'm sure attacking this is non-trivial, you damage your security posture by making source and destination traffic predictable across the Internet.
I've configured a good number of Application Gateways now for Enterprise applications and out of morbid curiosity I had a go at configuring a very basic one using HTTP to do what you're attempting - fortunately (yes, fortunately) I received an HTTP 502 so I'm going say that this isn't possible. I'll add that I'm glad it isn't possible because it's a Bad Idea (TM).
My suggestion is that you either migrate your DMZ servers to Azure (for the best performance/network latency) or implement a VPN or (preferably) ExpressRoute. You'll then be able to deploy an Application Gateway using the correct architecture where you terminate your users' connections at the App Gateway and that re-transmits the request within your RFC1918 network to your DMZ servers which respond within the network back to the App Gateway and ultimately back to the requestor.
Sorry it's not what you wanted to hear. If you're determined to do this, perhaps nginx could be made to?