Not an answer, only an idea to solve the problem.

I tried to solve your integral equation iterativ using NestList:

sol = NestList[
Function[fu,
FunctionInterpolation[
 1/3 (-2 Sqrt[1 - t] + 3 t - 4 t Sqrt[1 - t] - 4 t^(3/2)) + 
  NIntegrate[fu[s]/Sqrt[Sqrt[(t - s)^2]] , {s, 0, 1}, 
   Method -> "LocalAdaptive" ], {t, 0, 1 }]
] , 0 &,  (* initial function *)5];
Plot[Map[#[t] &, sol], {t, 0, 1}

Unfortunately the Picarditeration doesn't converge...

Perhaps you have additional system knowhow to force a convergent iteration?

Not an answer, only an idea to solve the problem.

I tried to solve your integral equation iterativ using NestList:

sol = NestList[
Function[fu,
FunctionInterpolation[
 1/3 (-2 Sqrt[1 - t] + 3 t - 4 t Sqrt[1 - t] - 4 t^(3/2)) + 
  NIntegrate[fu[s]/Sqrt[Sqrt[(t - s)^2]] , {s, 0, 1}, 
   Method -> "LocalAdaptive" ], {t, 0, 1 }]
] , 0 &,  (* initial function *)5];

Unfortunately the Picarditeration doesn't converge in your case:

    Plot[Map[#[t] &, sol], {t, 0, 1}

Perhaps you have additional system knowhow to force a convergent iteration?