Jelly, 2 bytes, score = 1

ḷḊ

Try it online!

A full program that takes a string and prints a string. When the first byte is removed, removes the first byte of the input.

Achieving a lower score would mean finding a solution where removing any byte leads to the corresponding byte being removed from the string, which I think will be challenging.

Explanation

ḷ  | Left argument of input string and:
 Ḋ | - Input string with first byte removed

With first byte removed:

Ḋ | Remove first byte