Just randomly thinking out loud: Is it possible to stop privately owned markets without the state having more control? I suppose the concern is with large private corporations, as opposed to smaller to medium sized businesses. I think that's what a lot of Democrats want to do, but I often associate them with bigger government. I know very little about politics though.

Sure, by enacting strict laws that forbid individual competitors from exceeding some amount of market share or forming conglomerates. Corporate breakups are self-executing. No compliance bureaucracy required, just investigators and lawyers. The government doesn't acquire the power to pick winners and losers, they indifferently breakup any player who gets big enough to distort the market or centralize too much power.

Of course it is. Private property in general is only possible because of enforcement by the state. Relaxing IP laws and other measures that limit the power of the state to enforce negative liberty would fit the bill.