Doesnt convexity prevent thick indifference curves aswell?
My approach would be to define thick indifference curves in terms of a stronger form of local non-satiation:


  Definition (thick indifference curves) Preferences are said to have thick indifference curves if there exists at least one bundle $A\in\mathbb{R}^l$ and an open ball
  $\mathscr{B}(A)$ around $A$ such that $A'\sim A$ for every
  $A'\in\mathscr{B}(A)$.


Let's set up a definition for strict convexity to be sure we are on the same page:


  Definition (strict convexity) Preferences obey strict convexity if
  
  $$A\sim B\implies \lambda A+(1-\lambda) B\succ A\sim B$$
  
  for $A,B\in\mathbb{R}^l$ and any $\lambda\in(0,1)$.


Now we can state the desired result:


  Proposition If $\succsim$ is strictly convex then $\succsim$ does not have thick indifference curves.


Proof Suppose that $\succsim$ has thick indifference curves. Then there exists an $A$ such that $A'\sim A$ for all $A'\in\mathscr{B}(A)$. Thus, fix two bundles $A$ and $B$ such that $A\sim B$ and $B\in\mathscr{B}(A)$. We know that 
$$\lambda A+(1-\lambda) B \in \mathscr{B}(A).$$
Thus, if the indifference curve is thick around point$A$ we must have $\lambda A+(1-\lambda) B\sim A\sim B$. But this contradicts the strict convexity hypothesis. QED



That should do the job for strict convexity. As frage_man already pointed out, the proposition fails to hold generally under weak convexity. Indifference everywhere is a very elegant example why.