The simple answer would be no. You can't register a T-shirt design as a trademark. However, it is possible to register a word, slogan, tagline or logo that you use on a T-shirt. More specifically, trademark Class 25 covers clothing and is certainly appropriate for protecting your trademark on clothing. Alternatively, a different type of IP protection, such as registered design, can also be applied to a specific t-shirt design.
In this case, it is certainly advisable to do a check of already registered trademarks in your market, given that many businesses are also seeking registration in the apparel class for their merch or promotional clothes. In fact, trademarks filed in this class are among the most numerous. Remember that for your registration to be successful, your mark needs to differ from existing marks in the same class and market.