We study Turing computable embeddings for various classes of linear orders. The concept of a Turing computable embedding (or tc-embedding for short) was developed by Calvert, Cummins, Knight, and Miller as an effective counterpart for Borel embeddings. We are focused on tc-embeddings for classes equipped with computable infinitary Σα equivalence, denoted by ∼c α. In this paper, we isolate a natural subclass of linear orders, denoted by WMB, such that (WMB, ≅) is not universal under tc-embeddings, but for any computable ordinal α ≥ 5, (WMB, ∼c α) is universal under tc-embeddings. Informally speaking, WMB is not tc-universal, but it becomes tc-universal if one imposes some natural restrictions on the effective complexity of the syntax. We also give a complete syntactic characterization for classes (K, ≅) that are Turing computably embeddable into some specific classes (C, ≅) of well-orders. This extends the similar result of Knight, Miller, and Vanden Boom for the class of all finite linear orders Cfin.