We exhibit examples of geometrically simple abelian surfaces A/ℚ with conductor bounded by (10 000)2 whose Tate—Shafarevich groups contain a subgroup isomorphic to (ℤ/pℤ)2 for each p = 5, 7, 11, 13. To find these examples we generalise work of Cremona–Freitas to enumerate all congruences of a certain type between pairs of weight 2 newforms f ∈ S2new(Γ0(N)) and g ∈ S2new(Γ0(M)) contained in the LMFDB (i.e., with N, M < 10 000) and with coefficient fields of degree ≤ 4. Passing from the modular forms to the corresponding abelian varieties we use visibility to (unconditionally) prove the existence of non-trivial elements of the Tate–Shafarevich group. Finally we construct an example of an abelian surface with (ℤ/7ℤ)2 ⊂ Ш(A/ℚ) which is (conjecturally) not visible in any abelian threefold.