This is a known problem for the GriGri, especially the newer version GriGri2. The GriGri2 is compatible with rope diameters of 8.9-11mm. The old one officially supports 10-11mm diameter, however is often used with thinner ropes as well (at your own risk). So the tendency for the rope to get stuck while feeding it out is certainly bigger with the new GriGri. However I manage to constantly jam the rope with an old GriGri and a 9.5mm rope as well.
This problem is one of the two major points for mishandling GriGris leading to dangerous situations (the other one is the lever not having a panic function). To give out quick slack you can often see people holding down the area where the lever is fixed (called cam). This disengages the blocking mechanism and is thus very dangerous. The official method for giving slack quickly is called "Gaswerk" method (after a climbing gym in Switzerland :) ). It is described here on the official Petzl site.
The single most important point is the same as always: You always keep hold of the braking strand. So the rope goes over your palm and all your fingers except your thumb go over it (tunnel). Then you use your thumb to press down the cam to disengage the blocking mechanism while paying slack and immediately remove the thumb when finished. You do not put any force onto the device with any other part of your hand/body. Especially you do not counteract pressing with the thumb on the top of the GriGri by pressing with another finger on the bottom. By following this advice you are safe at all times: If the leader falls while you pay out slack you only have the thumb on the knock which will slip away or at least not be able to exert enough force to keep the blocking mechanism disengaged (remember: there is nothing else on the device to counteract the thumb).
You should find someone experienced who can show you how to do this in real life. Then you need to practice, as it is not a completely intuitive thing to do. Some people pick it up near instantly, others struggle.
I myself keep using tubers most of the time for belaying a leader. Only for top-roping or checking out a project (hanging for long times) I use the Grigri, as it fully blocks the rope without you exerting any force. Of course you still keep the braking strand in your hand, but it need not be tensioned.