The linkage bound is 2.

If the algebraic group is simple, say over an algebraically closed $k$,
then one has the following lemma. 

Lemma. If $V$, $W$ are finite dimensional,
there is an $m$ depending on $V$, $W$, so that if  $St_n$ 
is the $n$-th Steinberg module with $n\geq m$ the natural map
 $Ext^i(V,W)\to Ext^i(V,W\otimes
St_n\otimes St_n)$ vanishes.

See for instance my 1977 
<A HREF="http://resolver.sub.uni-goettingen.de/purl?GDZPPN002093200">
joint paper with Cline, Parshall and Scott</A>. So this gives a way to kill extension classes while staying within
the category of finite dimensional representations.
Now it seems one may argue as in Sasha's answer, dualized.

Indeed consider a representative $E$ of an element of $Ext^1(V,W)$.
Let us write $W\otimes St_n \otimes St_n$ as $I_n(W)$.
For $n$ large the lemma implies that $E$ is the Yoneda composite $P\circ f$
of $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$ with an element 
$f:V\to  I_n(W)/W$. Now if one has a representative of $Ext^i(V,W)$ with $i>1$, 
write it as
Yoneda composite $E\circ F$ of an 
$E\in Ext^1(Z,W)$ and an $F\in Ext^{i-1}(V,Z)$. 
Applying the previous result one gets a
linkage map from $E\circ F$ towards a Yoneda composite $P\circ Q$ of the representative $P:0\to W\to I_n(W)\to I_n(W)/W\to 0$
with a representative $Q=f\circ F$ of an element of $Ext^{i-1}(V, I_n(W)/W)$. 
So we use that there is a linkage from $(P\circ f)\circ F$ towards 
$P\circ(f\circ F)$. 
One repeats this untill one has a linkage map from the original $E\circ F$
towards a Yoneda 
composite $R\circ S$ of an extension 
$R:0\to W\to I_{n_1}(W)\to I_{n_2}(X_1)\to \cdots I_{n_{i-2}}(X_{i-2})\to
I_{n_{i-2}}(X_{i-2})/X_{i-2}\to 0$ [ depending only on $W$ ] with an element $S$ of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$.
Given a second representative of the same class of $Ext^i(V,W)$ one may repeat the construction and if the $n_i$ are
big enough one ends up with the same class of $Ext^1(V,I_n(X_{i-2})/X_{i-2})$,
by a dimension shift argument. So the linkage is bounded by 2 as in 
Sasha's answer.

For reductive groups over an algebraically closed $k$ the Lemma still holds,
interpreted appropriately. For nonreductive groups one may take an exhaustive 
filtration $k\subset M_1\subset M_2\cdots$ of the coordinate ring $k[G]$ by finite dimensional submodules [ for the action by left translation ]
and replace $St_n \otimes St_n$ with $M_n$ in the argument above.
This is less explicit. It uses that the limit over $n$ of the $Ext^i(V,W\otimes M_n)$
vanishes.