Sounds like you need to strengthen your grip. The good news is that for most people, the static strength involved in grip develops pretty quickly. A few pointers:
1. Continue doing whatever exercises are challenging your grip. In my experience, building grip strength as a side effect of some exercise has been the most effective method. For example, the deadlift and pull-up increased my grip strength more than all the grip specific exercises I had done before that. If you need to lower the weight initially on the exercise, then check your ego and do it. It won't be long before your grip catches up and you can go back up in weight.
2. If #1 does not work after a few weeks, consider adding grip specific exercises. These include wrist curls, the wrist curler, and farmer walks.
3. Take care of the skin on your hands. Very often, grip strength is not always limited by muscle failure, but also by pain on the skin. Just like your muscles, your skin can adapt to the extra stress by becoming thicker and tougher and forming calluses. This simply takes time. However, you need to be wary of blisters. They are painful and can tear and seriously hamper your training. Make sure to take proper care of your hands by filing them down.
This page has a good tutorial.
4. Consider using chalk. This dries out your hands, helps prevent blisters and gives you a better grip.
5. Make sure you are holding the weight in the proper place. For most exercises, you want the bar you are holding to be as deep into your fingers as possible. Most people initially have the bar partially on their palm and the result is that a large chunk of the skin on the palm gets caught underneath. This is very painful and should be avoided. Instead, force the bar deeper into the fingers and avoid getting that skin flap under it. It will be more painful at first, but better in the long run.
6. Avoid straps or other artificial aids. Although they will help you hold much more weight, they will completely prevent you from developing grip strength. Grip strength is extremely useful in the real world and is necessary to allow you to apply most of your other strength in the real world, so take the time to develop it.