I have a 32" LG LCDTV (32LG40 I think is the model#) on W7 and it works fine, I don't remember ever having this issue. A few months ago I added an Onkyo 7.1 receiver to the mix and subsequently, I now have video signal going from my video card (ATI HD4850) in DVI to HDMI on the receiver and then HDMI out to the TV. Since adding the receiver I have had 1 or 2 instances in several hundred hours of operation in which the signal blacks out for a second or two but I have read that it's an intermittent issue with the Onkyo receivers on 1080p output (most commonly seen with PS3's).
What can you tell me about the cable you're using? Are you going native HDMI to HDMI or doing a DVI-to-HDMI thing like I am? It might actually be the cable you're using - I know you said it works in safe mode but that may be because safe mode is only trying to push a low resolution to the display. When I first hooked up a BD player to this TV I bought two HDMI cables - one was cheap (like $.90 on amazon) and one was like $10 on amazon. The cheap one blipped out and threw artifacts when I played blu rays in native 1080p - but played fine on 720. I suspect its cheap construction didn't allow for a clean 1080p signal, whether it was due to insufficient insulation quality or what I don't know, but the $10 one played 1080 or 720 just fine. It could be that your problem is similar - you have a cheap cable and in Normal mode W7 is trying to push a resolution (1650x1080 for instance) that the cable cannot transmit cleanly.
That would be the easiest solution to test - would be to go spend a little more money on an HDMI cable (or DVI to HDMI, w/e you're using), and see if it makes any difference. The only other possibility I can think of video drivers. For some reason your drivers don't like trying to push the max res that Normal Mode is trying to display. One way or another, I have a feeling that it's working in Safe Mode because it's not displaying in whatever high resolution or frame rate or something that it's attempting to use in normal mode.
Reply with video card specs, cable, driver version etc.