It depends on how you define big government. If you asked them if they support Medicare, Medicaid, and SS, I bet the majority of Americans of every age group will vote overwhelmingly for those programs. Paraphrasing Mark Steyn, "Americans want all the benefits of the European welfare state, but they don't what to pay European level taxes."
Demographically, our changing population fueled by mass immigration and minority birthrates, will increase the numbers of those wanting bigger government and more services. Almost one half of our population receive a government check in one form or another and that will increase as our population ages and the wealth gap widens creating a large, permanent underclass.
Once they get "robbed" so-to-speak, by either a real criminal or one representing the government, they'll drop the groupthink and realize they've been led by the pied piper of progressivism.
IMO we are going to see more and more people demanding more from the government. Even the Trump tax reform plan increases those who do not pay income taxes to about 50% of the population. 80% of Americans pay more in payroll taxes than they do in income taxes. Incomes have remained stagnant or declining since 1969 when measured in inflation adjusted dollars. We have the lowest labor participation rates since the 1970s. Mass immigration and automation are depressing wages and eliminating jobs.
Agree. The Government at the federal, state, and local levels control more than a quarter of our GDP.