Is the road where you take these corners bumpy? If your tires are bouncing when you hit bumps they may be overinflated. Check the cold inflation pressure against the recommended pressure printed on the placard located on the inside of the drivers door. Sometimes well meaning but uninformed tire and oil change shops will adjust the pressure to the maximum pressure listed on the tires. If the difference is great enough this may cause the tires to bounce over bumps especially when unloaded during cornering. As to the road grip, there are certainly tires that provide more than the original equipment but not usually without compromise. Super sticky summer only treads may impress with dry grip but will likely wear fast, produce more noise, and may not inspire confidence in the rain.