How can I prove my travel history if I've lost my passport?

Honesty I do not think it is possible and even a passport does not record all movements between any countries. Some countries do not use exit stamps, some no entry stamps and some do it on pieces of paper which you hold onto while there and hand back on exit.

Many countries keep their own records but those are only from a single point-of-view. For example, in Ecuador we con obtain our Cerficado Migratorio which is a list of all times a person has entered and exited the country. So, they know when I left and came back - even across passport changes - but they do not know from where or to where. Since my passport is Canadian, almost all lines in the report says that I arrived from Canada, although it does show the first immediate transit to another Andean Community nation, possibly because of some information accord. For example, when I traveled to Bolivia from Ecuador via Peru, my record says that I went to Peru. Apparently, they only share entry and not exit because, coming back on the same route it says that I came from Canada.

You would have to contact immigration offices of every country you visited to create a travel history. Even so, I am sure many countries do not keep records, do not keep them for longer than a certain period and may not have electronic record-keeping until some years ago, so would not be able to retrieve your records quickly, assuming they still exit.