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Manufacturers usually don't "market" designs, they just print them.
It's usually up to the designer to market and sell their designs.
You can sometimes try to approach larger established companies to see if they would like to purchase your designs, but that's like trying to approach a famous movie director with your untested script, it's not likely to get much response.
Another method would be to attend a tradeshow and showcase your designs there, and if there are interested buyers (for retail stores), then you can find a local screen printer to print hte shirts and you can sell wholesale to the retails stores. That way you're not printing any shirts before you have a buyer.
Most people who have great t-shirt ideas usually try to sell a few themselves. By either printing them themselves, outsourcing the printing to a local screen printer, or using a fulfillment service like cafepress.com,spreadshirt.com, zazzle.com, etc.
That's when you can "prove" your designs are interesting putting them up for actual people to buy. That is a good test of whether they are marketable. Then you can start working the wholesale route after that point.