Yes, you can indeed implement a Turing machine with DNA by modifying DNA strands to simulate Wang tiles; [0] this is modeled in the abstract self assembly model. [1] You can also build emojis with DNA. [2]
[0] https://thesis.library.caltech.edu/1866/1/winfree_thesis.pdf
[1] http://self-assembly.net/wiki/index.php?title=Abstract_Tile_...