How can I get edges to bend to avoid crossing?

Here's the one way by using the custom edge function:

arcRight[{a:{x1_,y1_},___,b:{x2_,y2_}}]/;y1>y2:=arcRight[{b, a}];
arcRight[{a:{x1_,y1_},___,b:{x2_,y2_}}]/;y1<=y2:=BSplineCurve[{a, {x1 + (y2-y1).7, (y1+y2)/2},b}]

iLayeredDrawing[g_, spos_Integer:1, opt___?OptionQ] :=
    Module[{s, vlist, leaves},
        vlist = VertexList[g];
        s = vlist[[spos]];
        leaves = MaximalBy[Reap[BreadthFirstScan[g, s, "DiscoverVertex"->(Sow[{#1,#3}]&)]][[2,1]], Last][[All,1]];
        Graph[vlist, EdgeList[g],
         opt, GraphLayout->{"LayeredEmbedding", "Orientation"->Left, "LeafDistance"->1/(Length[leaves]/2), "RootVertex" -> s}, EdgeShapeFunction->{a_\[UndirectedEdge]b_/;SubsetQ[leaves,{a,b}]:>(arcRight[#1]&)}]
    ]

For example,

iLayeredDrawing[PetersenGraph[], EdgeStyle -> Black, 
 VertexStyle -> Directive[White, EdgeForm[Black]], VertexSize -> .3]

iLayeredDrawing[FromEntity[Entity["Graph", "HoffmanSingletonGraph"]], 
 EdgeStyle -> Black, 
 VertexStyle -> Directive[White, EdgeForm[Black]], VertexSize -> .6, 
 ImageSize -> 600]

1. Rescale the vertex coordinates given by "LayeredEmbedding" to run form 0 to 1 in each dimension,
2. Pick the edges with vertices in the right-most layer by checking if both vertices have first coordinate equal to 1.,
3. Use a slightly modified version of the built-in edge shape function "CurvedArc" as the EdgeShapeFunction for the right-most edges.

ClearAll[eSF, toLayered, layeredG]
eSF[curv_: 1] := GraphElementData[{"CurvedArc", "Curvature" -> curv}][
    SortBy[-Last @ # &] @ #[[{1, -1}]], ##2] &;

toLayered = Module[{el = EdgeList[#], g, re, 
     vcoords = Round[#, .001] & @ Transpose[Rescale /@ 
        Transpose[GraphEmbedding[#, {"LayeredEmbedding", "Orientation" -> Left}]]]},
    g = SetProperty[#, VertexCoordinates -> vcoords];
    re = Pick[el, First[PropertyValue[{g, #}, VertexCoordinates]]& /@ Apply[List][#] == 
        {1., 1.}& /@ el];
    SetProperty[g, { EdgeShapeFunction -> {Alternatives @@ re -> eSF[]}}]] &;

We compose toLayered with Graph to get a function that takes the same arguments and options as Graph:

layeredG = toLayered @* Graph;

Examples:

layeredG[PetersenGraph[], EdgeStyle -> Black, 
 VertexStyle -> Directive[White, EdgeForm[Black]], VertexSize -> .3]

hsg = FromEntity[Entity["Graph", "HoffmanSingletonGraph"]];
layeredG[hsg, ImageSize -> 600, VertexSize -> .8, 
  VertexStyle -> White, EdgeStyle -> Gray]