No, physicists do not think that space is like a tinily subdivided grid. Lots of physics would break if that were the case, including QM.

We have a lot of hints that the amount of information in any given space might be bounded, but yet that space appears continuous. How exactly you reconcile this is (one of) the mystery of quantum gravity.

Is there a good write-up somewhere of the problems with discretization of space for physics?

Interestingly, some great mathematicians (including Grothendieck) thought that modelling space as continuum was an approximation and not the reality.