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?
