How to ask Mathematica to fill in colors between curves in the given code?

ListPlot[
 {Thread[{x, y}], Thread[{x, y^2}],
  Thread[{x, y^3}], Thread[{x, y^4}]},
 Filling -> {
   1 -> {{2}, Opacity[0.3, Red]}, 
   3 -> {{4}, Opacity[0.3, Blue]}
 },
 PlotStyle -> {Red, Red, Blue, Blue}, Joined -> True,
 PlotRange -> {1, 3}
]

You can also construct a TemporalData object from your x and y and use it with ListLinePlot:

td = TemporalData[Table[y^i, {i, 4}], {x}];

llp = ListLinePlot[td,
 PlotStyle -> {Red, Red, RGBColor[.45, .77, .92], RGBColor[.45, .77, .92]}, 
 Filling -> {1 -> {2}, 3 -> {4}}, 
 PlotRange -> {1, All}]

Alternatively, use Table[y^i, {i, 4}] as input data and add the option DataRange -> MinMax[x]:

llp = ListLinePlot[Table[y^i, {i, 4}], 
 DataRange -> MinMax[x],
 PlotStyle -> {Red, Red, RGBColor[.45, .77, .92], RGBColor[.45, .77, .92]}, 
 Filling -> {1 -> {2}, 3 -> {4}}, 
 PlotRange -> {1, All}]

SeedRandom[1]
Normal[llp] /. p_Polygon :> Dynamic[texturedPoly[CurrentValue["Color"]][p]]

where

ClearAll[texturedPoly]
texturedPoly[color_: Red, thickness_: 10, n_: 200, sd_: 100, jitter_: 5][p_Polygon] := 
 Module[{BSF = BSplineFunction[Join @@ (RandomSample[Partition[#, 20]] & @ 
   SortBy[RandomPoint[BoundingRegion[p[[1]], "MinConvexPolygon"], n], First])]}, 
  {Opacity[.8], Texture @ 
   ImageEffect[Graphics[{CapForm["Round"], JoinForm["Round"], 
      color, AbsoluteThickness[thickness], 
      Line[BSF /@ Subdivide[sd]]}], {"Jitter", jitter}], 
   Polygon[p[[1]], VertexTextureCoordinates -> Automatic]}]