I'm not entirely sure what you're asking, so if I misinterpret you, please let me know.

No, attackers do not always suffer more casualties than defenders - frequently they suffer far less, as in the German invasion of the Soviet Union. D-Day was a massive Allied success and a critical German failure, owing in large part to the concentration of superior Allied forces against relatively weak German defenses - it's what happens when you try to defend 2,000 kilometers worth of seacoast. It is worth remembering that four of the five invasion beaches were taken with relatively little trouble and relatively few Allied casualties. Omaha, the only beach that turned into a genuine bloodbath, was unique in geography, defending troop strength, and artillery support, and the Americans still took it despite suffering ~2,000 casualties. Local German counter-attacks launched on the invasion beaches late on June 6 failed badly and with high casualties. While it's certainly true that the invasion bogged down in subsequent days and turned into two months of bitter attritional warfare, June 6 largely went according to plan.