Let k be an algebraically closed field, G a linear algebraic group over k and φ ∈ Aut(G), the group of all algebraic group automorphisms of G. Two elements x; y of G are said to be φ-twisted conjugate if y = gxφ(g)–1 for some g ∈ G. In this paper we prove that for a connected non-solvable linear algebraic group G over k, the number of its φ-twisted conjugacy classes is infinite for every φ ∈ Aut(G).