The more cities you have, the more research costs go up, so it's not uncommon to fall behind. If you want to stay competitive, you have to start micromanaging and building up cities to be commerce/research powerhouses, using great persons and the right improvements/buildings to emphasize them, with those cities the using their hammers to build research or gold as necessary.
Looking at your image, the cities I see are mostly about food and hammers, not commerce.
You also want keep to the civics that allow you to best deal with quantity of cities. I usually keep to Despotism, Merchant Princes, and Plutocracy for as long as possible.
And I keep from expanding constantly. I try to go through alternating waves of expansion and building up my cities. During expansion my research my drop to as low as 35%, but I try to get it back to ~70% before I return to expanding.
Oh, and learn to play tall, not wide. Avoid having many cities, especially with overlapping territories. Settle cities to maximize their purpose (research/gold/units) and don't be afraid to have gaps between cities of wasted tiles, or about trying to get every single resource. Focus on getting the resources that city specifically needs to achieve its goals.