It sounds like you have developed tendonitis, specifically, golf elbow. This is one of the most common injuries I see in the sport and am no stranger to experiencing it. myself. I'm more prone to tennis elbow, which is the same thing, but on the top side of the forearm. The hand is an amazing instrument. It is powerful yet designed to perform intricate tasks. mostly everything that powers the hand is in the forearm. This way, it can be strong but without bulkiness. When you over use your grip, you will feel it in the tendons of your forearm. Tennis/golf elbow usually starts up toward the elbow where the tendons attach. Once they are over worked, they become inflamed. As this inflamation gets worse, more of the tendon will become inflamed and you will feel pain traveling further down the tendon toward the hand. Conversely, as you begin the healing process, you'll notice the pain will slowly retreat up the forearm, back to its source. Healing this can take quite some time. Tendons get very poor blood flow, so it is hard for fresh blood to get into the damaged areas and carry away waste, especially when the tendon is inflamed. Inflamation is like a traffic jam, making poor circulation even worse. Fixing this is going to take patience and a depending how bad it has gotten, a good deal of time. If you are hardly able to grip a bar, I would suggest you start by taking 1 full wk off of training. I know, I know. You don't have to explain how awful that sounds, but if you don't get aggressive with rehabbing this, it isn't going to get better.
When you come back, you need to be aware of what is causing the most issue. Anytime you grip the bar tightly, it's going to make it worse, so use wraps. For now, I would use machines, so you dont have to grip the bar when pushing and dont have to stable the bar when pulling.
REST
ICE directly on the area, 20 min on for as many times a day as you can.
NSAID I use motrin 800
the rest, ice and drug will help to get the inflamation under control.
MASSAGE will help to bring new blood into the tendon and break up scar tissue
REHAB, find exercises on youtube that you can do a couple times a day to strengthen your grip strength so that you can avoid this from happening again. there is a good chance that this issue will pop up again in the future, so pay attention and if you catch it early, you can use these tools to reverse it in the same wk.