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}

[![enter image description here][1]][1]

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

  [1]: https://i.sstatic.net/UjgVy.png