This is what I figured out to adjust Singular's printlevel:
from sage.libs.singular import function_factory
execute = function_factory.ff.execute
execute('printlevel=-1')
Now that previous example runs without all the clutter:
sage: R.<x,y> = GF(3)[]
....: S = R.quo(ideal(y^4 + y - x^5))
....: from sage.libs.singular.function import singular_function
....: normalC = sage.libs.singular.function_factory.ff.normal__lib.normalC
....: normalC(S.defining_ideal(), ['withGens'])[1][0]
....:
[y^4 + y, x^4*y, x^3*y^3 - x^3*y^2 + x^3*y, x^2*y^3 - x^2*y^2 + x^2*y, x^3*y
^2 + x^3*y, x*y^3 - x*y^2 + x*y, x^4]
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