How does treewidth behave under graph minor operations? It is a well-known fact that for any minor H of a graph G (commonly written as $H \leq_m G$), the treewidth of H is smaller than or equal to that of G.

Minors of a graph are created through the operations of (1) vertex deletion, (2) edge deletion and (3) edge contraction. I am curious as to whether one can bound the decrease in treewidth when applying any single operation from (1)-(3). 
Of particular interest for me is the question whether said decrease can always be bounded by some constant.
The decrease in tree width of any of the three operations will be 0 or 1.  The proof is simple (I will take the case of the vertex deletion).

Let $G$ be the original graph and $G' = G - v$ be the graph obtained by deleting $v$.

Let $T'$ be an optimal tree decomposition of $G'$, with $\text{tw}(G') = w'$.

Now, construct $T$ from $T'$ by adding $v$ to every bag of $T'$.  Clearly, $T$ is a tree decomposition for $G$ with width $\text{tw}(G) \leq \text{tw}(T) = 1 + \text{tw}(G') = 1 + w'$.