The current state of the UK water industry doesn't seem to support this theory. Privatisation has only lead to water companies like Thames Water taking on unsustainable debt while paying out billions in dividends, underinvesting in infrastructure, and polluting like crazy. Now they are demanding permission from the regulator to massively hike prices, because the foreign investment funds that own them are apparently unwilling to countenance the idea of losing any money on their investments.
Infrastructure like public transport and utilities are not, and never will be, functional markets, and regulation is always captured or ineffective in the long term. Privatisation is only a method to let financial markets pillage public goods.