Has there been a game where vulkan performance has been better then dx12? Whenever they are side by side vulkan always performs worse in my experience.

There have been cases like that, e.g., RDR2, but I think it mostly comes down to implementation quality.

If games run faster on linux with DX12 translated to Vulkan than they do on identical hardware running on Windows 11 then I can't imagine a particularly big performance difference.