I have had this problem and unfortunately never found a good solution. Video tuners are just not made for video games. They have an inherent lag due to the conversion (and in some cases compression) that the card does. In some (like the $200 one I bought) this lag was up to 3-4 seconds and makes the game completely un-playable. I returned that one and went back to a $60 USB one that has worse quality, but the software does less work and so the lag is cut to just a fracton of a second. In a game like GT4 where precise control is necessary, this is still unacceptable.
The idea was to not have to buy a TV for my dorm room. The tuner worked fine for watching TV, since it doesn't really matter if there is a lag, but when I got GT4 I wound up having to buy a 14-inch-er anyway. I now only use the video card to capture replays, when real-time control is not crucial.
If you find a solution to this problem, or a video card that does not suffer from it, please let us know!