What evidence do we use that 2nd wave

2nd wave or we opened too soon

For a second wave, usually there would be the need for a trough after the first wave.  It looks to me that initial lockdown started to produce some results, but the improvement was slow. It looks as if the ending of the lockdown came too soon (that is the virus was still widely circulating in the community allowing a rapid rise).  

The UK has been harder hit that the USA when considering deaths per population (UK 66/100,000; USA 38/100,000 as of end of June), but the UK seems to be following more of a wave pattern

My personal views are that apart from political problems to a lot of the response to this virus, the USA has structural problems (very expensive health system that does not cover the whole population, few safe guards to help the unemployed) that have forced the country to open up too fast.  These structural problems are the result of choices made over decades, and unfortunately they are causing problems now that will do great damage to the USA.  

Interestingly, although I don't have a graph for it to show you, it seems that Canada has followed a more wave like path than the States has done (its recent numbers are quite low).  Also the death rate in Canada is much lower (23/100,000), despite having a very long shared boundary.