MySQL database queries start timing out after months of inserts/deletes

I have a MySQL database (innodb) that has been running for about a year. It has consistently been inserting and deleting from 2 specific tables in that row continuously for that time, keeping at a relatively consistent number of rows.

I'm finding that my queries against these tables are periodically timing out now, and it's not clear why that would be the case (nor how to resolve it). The queries previously worked flawlessly at that scale, so it seems like the tables might have become fragmented.

Why have reads/writes/deletes slowed down to the point of timing out, and how do I improve that behaviour?