Perhaps we have different ideas in mind with the terms "public" and "private". The power of both the government (i.e., "public") and corporations (i.e., the business part of the "private" sector) are interdependent, and they wax and wane together.
When we citizens complain about a problem, asking the government to stop it, the legislature makes laws intended (if we're feeling charitable) to rein in the actions of industry. This is done by creating regulatory agencies. But who to staff those agencies with? We need regulators who actually understand what the industry does. Thus, the regulators wind up being staffed by people from the industry, or at least people having a mental paradigm matching that of the status quo. This is called "regulatory capture", and the effect is that over time, the forces intended to bind industry winds up reinforcing the industry.
So what we see is that giving the government more power, intentioned to protect us, winds up subverted and perverted to protecting the industries that were in the crosshairs. This necessarily happens, despite best intentions; see "public choice economics". The more you empower government with the intention of controlling corporations, the more power eventually winds up being wielded by those captured regulatory agencies, helping the corporations.
And the result is that our system today is frequently referred to as "corporatist", meaning that the power of government serves to keep industry afloat.
Corporations wield much more power today, yes. But that's only because we've been suckered into giving more power to the government (ostensibly to control industry). So the power increases in lockstep, and if you want to decrease the power of industry, you need to strip the government power that they've captured. Weaken the corporations by weakening the government, in other words.