What I like about NGrams is they're great for idiomatic preference "beauty contests"...

But thesis has two common meanings. Both the big hitters in that chart (submit, present) are mainly for the "university degree component" sense. So if we exclude them we can focus on the remaining alternatives - which are far more likely to occur in general conversation/debate...

The three clear front-runners there (advance, put forward, argue) all suggest "combative" championing of a position, which is what we'd expect in the non-academic context (where you're more likely to be quizzed and even challenged back by sceptics, in a debate).
I assume everyone agrees academia accounts for most of the present, submit usages. The main point of the pretty pictures is to show we use lots of different verbs in more everyday contexts (I could have added expound, introduce, etc., but it would just become unreadable).
TL;DR: You need to push your thesis forward a bit, but the exact word doesn't really matter.