We see our face when the light from the reflector  reaches our eye to form an image. But in moving water the light rays are reflected back in multiple directions due to turbulence and uneven surfaces . Hence an image is not formed. So we can not see our face.

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If it was still water instead of moving water we could have seen the reflection, but as the water is moving the reflected rays are thrown in multiple direction instead of meeting our eyes, due to fast motion and rough surface, hence we don't see any reflection.
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