A　digraph　D　of　order　n　is　k-traceable　if　n≥k　and　every　induced　subdigraph　of　D　of　order　k　is　traceable.　An　oriented　graph　is　a　digraph　without　directed　cycles　of　length　two.　The　Traceability　Conjecture　(van　Aardt　et　al.　(2008)　[6])　states　that　any　k-traceable　oriented　graph　of　order　at　least　2k−1　is　traceable　for　k≥2.　In　this　paper,　we　study　the　Traceability　Conjecture　under　forbidden　subdigraphs　conditions.　We　firstly　verify　that　any　k-traceable　oriented　graph　D　without　directed　cycles　of　length　4　but　containing　induced　directed　cycles　of　length　at　least　5　has　order　at　most　2k−3.　This　strengthens　a　result　of　van　Aardt　et　al.　(2011)　[5]　as　well　as　a　result　of　Lichiardopol　(2016)　[10].　We　also　show　that　every　k-traceable　oriented　graph　of　order　n≥3k−3　without　two　directed　triangles　mutually　sharing　a　unique　common　arc　is　traceable,　which　in　a　sense　extends　another　result　of　Lichiardopol.　©　2023　Elsevier　B.V.