Currently I am using an EF-S 18-200mm f/3.5-5.6 IS. But I feel this range is not sufficient for me, shooting events and sports. So I am considering an EF 28-300mm f/3.5-5.6L IS USM. It has a bit more reach (300 vs 200). Can you advise if it would give me clearer pictures and be more suitable for events and sports? I am using a Canon 650D.
The way to improve your image quality is to select a lens with a narrower range of focal lengths. In a lens like the EF-S 18-200mm f/3.5-5.6 IS there is such a wide range of focal lengths that too many compromises have to be made in the design of the lens. That is why you see professional sports photographers with 2 or 3 camera bodies: They have a short lens mounted on one and a long lens mounted on another. My two body set up usually includes either a 17-40mm or 24-105mm mounted on my FF body and a 70-200mm mounted on my crop body.
If you compare the EF-S 18-200mm f/3.5-5.6IS to the EF 28-300mm f/3.5-5.6 L IS at The-Digital-Picture, your current 18-200 is sharper than the 28-300 at the focal lengths and apertures they share in common when both are mounted on an APS-C body. The 28-300 tests better mounted on a full frame camera than on an APS-C body. If you do the same comparison at DxO Mark, the 28-300mm does a little better compared to the 18-200, but the EF 70-200mm f/2.8 L IS II blows both of them away. At 70mm it is significantly sharper at f/2.8 than either of the other lenses at f/5! At 200mm it is sharper at f/2.8 than the other lenses at f/5.6, although the 28-300 comes close in the center. If you look at the 'measurements-->sharpness-->profiles tab and set each lens to 200mm @ f/5.6 you can see that the 28-300mm is slightly sharper in the center than the 18-200mm, but just as soft on the edges as the 18-200mm. On your 600D body, you will see very modest image quality improvements going from the EF-S 18-200mm f/3.5-5.6 IS to the EF 28-300mm f/3.5-5.6 L IS, mainly in terms of edge/corner sharpness in the wider focal lengths. You will see a significant improvement by going to a lens such as the EF 70-200mm f/4 L or the EF 70-200mm f/2.8 L IS II.
Go to a sporting event and have a look at all the pro photographers out there shooting. I do not know of a single one that shoots with a Zoom as their primary. I do see many of them with more than one body sitting on PRIMES like 400mm or 600mm. The reason why a prime is that you will get far better image clarity with it than with any zoom. That being said the reason why you want the multiple bodies is that one of them will have a zoom with a top end of 200 or 300mm for close work. I specialized on portrait work and my lens of choice was the Canon 85mm f1.4, this was my prime lens. I so wanted the 1.2 but that was way out of my price range at the time.
It would also help to know what type of events you will be shooting and what vantage point you will have while shooting. If you are on the field then get a 400mm and a monopod and another body for your existing zoom.