The　Turán　number　of　a　graph　H,　denoted　by　ex(n,　H),　is　the　maximum　number　of　edges　in　an　n-vertex　graph　that　does　not　contain　H　as　a　subgraph.　For　a　vertex　v　and　a　multi-set　F　of　graphs,　the　suspension　F　+　v　of　F　is　the　graph　obtained　by　connecting　the　vertex　v　to　all　vertices　of　F　for　each　F　∈　F.　For　two　integers　k　≥　1　and　r　≥　2,　let　Hi　be　a　graph　containing　a　critical　edge　with　chromatic　number　r　for　any　i　∈　{1,　…,　k},　and　let　H　=　{H1,　…,　Hk}　+　v.　In　this　paper,　we　determine　ex(n,　H)　and　characterize　all　the　extremal　graphs　for　sufficiently　large　n.　This　generalizes　a　result　of　Chen,　Gould,　Pfender　and　Wei　on　intersecting　cliques.　©　The　authors.