A　digraph　D　is　r-free　if　such　D　has　no　directed　cycles　of　length　at　most　r,　where　r　is　a　positive　integer.　In　1980,　Bermond　et　al.　showed　that　if　D　is　an　r-free　strong　digraph　of　order　n,　then　the　size　of　D　is　at　most　[Formula　presented]　and　the　upper　bound　is　tight,　for　r≥2.　Namely,　they　determined　the　Turán　number　of　Cr-free　strong　digraphs　of　order　n,　where　Cr={C2,C3,…,Cr}　and　Ci　is　a　directed　cycle　of　length　i∈{2,3,…,r}.　Specially,　for　r=3,　the　maximum　size　of　3-free　strong　digraphs　of　order　n　is　[Formula　presented].　Let　Φn(ξ,γ)　be　a　family　of　3-free　strong　digraphs　of　order　n　in　which　the　minimum　out-degree　is　at　least　ξ　and　the　minimum　in-degree　is　at　least　γ,　where　both　ξ　and　γ　are　positive　integers.　Let　φn(ξ,γ)　be　the　maximum　size　of　digraphs　of　Φn(ξ,γ),　i.e.,　Turán　number　of　3-free　strong　digraphs　of　order　n　with　out　and　in-degree　restrictions.　We　denote　ϕn(ξ,γ)={D∈Φn(ξ,γ):thesizeofDisequaltoφn(ξ,γ)}.　Recently,　Chen　et　al.　described　ϕn(1,1),　i.e.,　all　3-free　strong　digraphs　of　order　n　with　size　[Formula　presented].　They　also　gave　the　bound　of　φn(2,1),　that　is,　[Formula　presented].　In　this　paper,　we　improve　the　upper　bound　of　φn(2,1)　to　[Formula　presented]　and　we　thus　get　[Formula　presented].　In　addition,　we　show　that　[Formula　presented]　by　constructing　two　3-free　strong　digraphs　with　minimum　out-degree　two　whose　size　reaches　[Formula　presented]　for　n=7,8,　and　verify　[Formula　presented]　for　n=9.　As　a　consequence,　Turán　number　of　3-free　strong　digraphs　of　order　n　with　out-degree　at　least　two,　i.e.,　φn(2,1),　is　one　of　[Formula　presented]　and　[Formula　presented].　©　2022　Elsevier　B.V.