For a simple $H$ with max degree $\leq$ 3, show a simple graph $G$ contains a subdivision of $H$ $\iff$ $G$ contains $H$ as a minor $\Rightarrow$ Is quite easy, since if $G$ contains a subdivision $G'$ of $H$ as a subgraph, then each $\{u,v\}$ edge in $H$ is represented by a path $P=\{u, ... v\}$.
Some of these paths might have length 1, but at least one of them has length $\geq 2$ by definition of a subdivision.
So if there are $k$ paths of length $\geq 2$, then $\forall i, 1 \leq i \leq k$,  $P_i=\{v^i_1\, e^i_2, v^i_2, e^i_3, ..., e^i_t, v^i_t\}$ s.t. $e_i=\{v^i_1, v^i_t\} \in H$.
Then we can just contract $P_i$ into $e_i$ by having the sequence of contractions $H_2,H_3,...,H_t$ such that $H_j$ is the contraction of edge $e_j=\{v_{j-1}^i, v_j^i\}$ into the vertex $v_j^i$.
If we do this for each edge, we obtain $H$, and since we start from $G' \subset G$, it means $G$ contains $H$ as a minor.  
However, I am not sure how to proceed for $\Leftarrow$.
I think we need to use the fact that each vertex has degree $\leq 3$. If $G$ contains $H$ as a minor, then some $k$ vertices $v_1, ..., v_k$ , $1 \geq k$, have been obtained by a series of contractions. For some vertex $v_i$ that was contracted, then $H_0^, ..., H_t$ is the contraction of edges $e_0, ..., e_t$ such that $e_o \in G$ and $e_t$ has $v_i$ as an endpoint.
I think we then need to consider the degree of $v_i$ (either 1, 2 or 3), which limits the degrees of the endpoints of the deges that were contracted, but I am not sure where to go from there.
Could someone help me find the right approach for $\Leftarrow$?
Thanks!

If $G$ contains $H$ as a minor, then $G$ can be reduced to $H$ after a series of deletions of vertices and edges, and contractions of edges. What you need to show is that the contraction of an edge in this case can itself be replaced by a series of deletions of vertices and edges, and suppressions of vertices.
So, consider the contraction of an edge $e=\{x,y\}$. 
1st case: At least one of $x$ and $y$ have degree 1, let's say $x$. In this case, contracting $e$ is essentially the same as deleting $x$.
2nd case: Both $x$ and $y$ have degree 2. In this case, contracting $e$ is essentially the same as suppressing either one of $x$ or $y$.
3rd case: Both $x$ and $y$ have degree 3 and a common neighbour $z$. In this case, contracting $e$ is essentially the same as deleting $\{y,z\}$ and then suppressing $y$. 
4th case: Both $x$ and $y$ have degree 3 and do not have a common neighbour. In this case, contracting $e$ would give as a vertex of degree 4, a contradiction.
Therefore, you can always replace the contraction of an edge, in the case where $H$ has maximum degree at most 3, with deletions of vertices and edges, and suppressions of vertices, which means that if $H$ is a minor of $G$, then it is a subdivision of $G$.
