Yes, in principle you will have higher shear stress in turbulent than in laminar flow. But without an explanation as to why this is so, it is like tossing a coin in the air ;).

For flow in a pipe (see http://www.efm.leeds.ac.uk/CIVE/CIVE1400/Section4/laminar_turbulent.htm), the shear stress depends on the flow speed and so does the Reynolds number. Hence, the shear stress is larger  in the turbulent flow than that in the laminar flow. It is not clear to me if this is always the case though.

The local shear stress on a wall at high Reynolds numbers depends on the normal gradient of the tangential velocity (dU/dy) in a layer near the wall called the viscous (sub)-layer. This gradient depends on the velocity profile of the boundary layer. For a laminar boundary layer the length scale associated with the boundary layer thickness is inversely proportional to the square root of the Reynolds number (1/sqrt(Re)). With increasing Reynolds number the wall shear stress in a laminar flow increases monotonously. At a certain moment the flow becomes unstable and turbulence appears. Turbulence involves eddy motion that transfers momentum much more efficiently than molecular momentum transfer, which dominates in laminar flows. This can be described in terms of a so-called turbulent viscosity. This viscosity reduces as one approaches the wall because the size of the vortices that can occur is limited by the vicinity of the wall. Prandtl assumes that this size decreases linearly as we approach the wall. Very near the wall the turbulent viscosity vanishes and the flow is again dominated by viscous momentum transfer. However the vortices in the outer flow have dramatically changed the global boundary layer velocity profile  U(y). The velocity far from the wall  (at the edge of the boundary layer) is much more uniform and this high velocity is found much closer to the wall than in a laminar flow. The result is a very steep gradient near the wall (large values of dU/dy) and a high wall shear stress.

Typically the velocity profile U(y) is linear in a laminar boundary layer while is is proportional to y^(1/7) in a turbulent boundary layer (outside the viscous dominated sub-layer). Using the y^(1/7) law in a so called Von-Karman boundary layer equations (integral formulation) yields quite reasonable order of magnitude of the evolution of turbulent boundary layers. 

Note that the resistance of objects to the flow has a wall shear stress component (viscous drag) plus a  pressure difference component) pressure drag). The transition from laminar to turbulent flow around a ball (or a cylinder in cross-flow) induces a reduction of the pressure drag, which is much larger than the increase in viscous drag. This results in a turbulence induced drag reduction. This effect is related to the delay in the flow separation of the boundary layers. This delay is due to the improved momentum transfer by turbulence within the boundary layer. The fluid floe to the wall is dragged up the adverse pressure gradient by the eddy viscosity!

Indeed a superficial analysis will often lead to wrong answers.

 depends on the definition.

sometimes the turbulence can decrease the resistence

Turbulence reduces the pressure drag in some cases but always increases the wall shear stress and hence the viscous drag.

The stronger dependence of pressure drop on mass flow rate for turbulent flow results from the fact that more energy has to be supplied to maintain the violent eddy motion in the fluid.