At the moment, Shopify does not have the functionality to charge an extra shipping fee for additional items in the cart. Thus only first product shipping charges are applied. This is only the case when products with the same shipping profile are ordered. For example, if products from different groups are in the cart (e.g. t-shirt and hoodie), the flat rate will be calculated correctly: first item shipping price (group1) + first item shipping price (group2).
As an alternative, you can use Printful live rates. You can then manually set a higher flat rate for the shipping profile groups that are often ordered at your store, or use third-party apps like Advanced Shipping Rules to set up additional item shipping fees for flat rates.