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• Does not depend on PlanarFaceList and we can distinct the boundary lines.

Clear["Global`*"];
pts1 = {{0, 0}, {1, 1}, {2, -1}, {3, 0}, {5, 2}, {6, -1}, {7, 3}};
pts2 = {{5, -2}, {0, 0}, {1, 2}, {5, 1}, {2, 2}, {4, 3}, {5, 4}};
pts3 = {{4, 2}, {3, 2}, {2, 3}, {-1, -3}};
curves = {curve1, curve2, curve3} = 
   BezierCurve /@ {pts1, pts2, pts3};
g = Graphics[{Arrowheads[.02], Arrow /@ curves}];

lines = MeshPrimitives[DiscretizeGraphics@#, 1] & /@ curves;
data = Region`Mesh`SplitIntersectingSegments[lines];
pts = data[[1]];
splits = data[[2]];
intersections = Cases[splits, l_ /; Length@l == 3, -1];
boundaryLinesIndexs = 
  Join[{#[[1, 2]]}, Range[#[[1, -1]], #[[2, 1]]], {#[[2, 2]]}] & /@ 
   Partition[intersections, 2];
reg = BoundaryMeshRegion[pts, Map[Line, boundaryLinesIndexs, {1}]];
Graphics[{{HatchFilling[], reg}, curves, Arrowheads[{{Large, .5}}], 
  Thread[{{Red, Green, Blue}, 
    Map[Arrow, pts[[#]] & /@ boundaryLinesIndexs, {1}]}]}]

• https://mathematica.stackexchange.com/a/295351/72111 still work for this case, but it can not distinct the three boundary lines.

Clear["Global`*"];
pts1 = {{0, 0}, {1, 1}, {2, -1}, {3, 0}, {5, 2}, {6, -1}, {7, 3}};
pts2 = {{5, -2}, {0, 0}, {1, 2}, {5, 1}, {2, 2}, {4, 3}, {5, 4}};
pts3 = {{4, 2}, {3, 2}, {2, 3}, {-1, -3}};
curves = {curve1, curve2, curve3} = 
   BezierCurve /@ {pts1, pts2, pts3};
g = Graphics[{Arrowheads[.02], Arrow /@ curves}];

lines = MeshPrimitives[DiscretizeGraphics@curves, 1];
data = Region`Mesh`SplitIntersectingSegments[lines];
pts = data[[1]];
splits = data[[2]];
segments = Flatten[Partition[#, 2, 1] & /@ splits, 1];
graph = Graph[Range@Length@pts, UndirectedEdge @@@ segments, 
   VertexCoordinates -> pts];
faces = PlanarFaceList[graph];
polys = Polygon[pts[[#]]] & /@ faces;
Show[g, Graphics[{HatchFilling[], EdgeForm[{Thick, Red}], 
   polys[[2]]}]]