Treewidth of deep Sierpiński Sieve Graph
You can recursively decompose each triangle into a smaller triangle and a trapezoid, and each trapezoid into two smaller triangles. The corresponding tree decomposition (whose bags contain the corners of a triangle at one level of the decomposition and a triangle or trapezoid at the next or previous level) has five vertices per bag and therefore has width four.

There is no $K_5$ minor, because the graph is planar. There is, however, a $K_{2,2,2}$ (octahedron) minor (actually, a subdivision), shown below. As this is one of the forbidden minors for treewidth three, the width is four.