The sentence "Barbed wire was invented in Illinois it quickly became popular all over the US" is a run-on sentence (specifically, a fused sentence)—not a fragment.
It joins two independent clauses with no punctuation or conjunction:
Option 1 (Period): "Barbed wire was invented in Illinois. It quickly became popular all over the US."
Option 2 (Semicolon): "Barbed wire was invented in Illinois; it quickly became popular all over the US."
Option 3 (Comma + conjunction): "Barbed wire was invented in Illinois, and it quickly became popular all over the US."
Option 4 (Subordination): "After barbed wire was invented in Illinois, it quickly became popular all over the US."
A fragment lacks a subject or verb (e.g., "Invented in Illinois." or "Barbed wire in Illinois."). This sentence has two subjects and two verbs—the problem is too much sentence, not too little.