How to find currents in a mesh system with only current sources?
Redraw the circuit as:






simulate this circuit &ndash; Schematic created using CircuitLab

On further simplification:







simulate this circuit

if R = \$\frac{(R_2\times R_3)}{(R_2+R_3)} = 12 \Omega \$,
then using current divider formula:

$$I_1 = -I\times \frac{R}{(R+R1)} = -10 A$$
Similarly you will get \$I_5 = +12A\$ and therefore \$I_2 = -30 - I_1 + I_5 = -8 A\$
From the circuit its clear that, by KCL:
$$I_2 = I_4 - I_3$$
Using current divider formula again, we will get
$$I_4 = I_2\times \frac{8}{24+8} = -2 A $$
therefore \$I_3 = -2 + 8 = +6 A\$

I think I know where you went wrong.

Source transformation:

Consider this circuit:







simulate this circuit

Its obvious that current through R1 = 4A and R2 = 6A

Suppose you transformed the current source with R1, across points A and B as follows:







simulate this circuit

Using this resulting circuit, you can calculate only the current through the external circuit of the equivalent voltage source**, i.e., whatever is at the right side of A and B. You can calculate the current across R2, $$ I = 60/10 = 6 A$$ Which is the same as we obtained before in the original circuit. 

But if you try to calculate the current through R1, which is internal to the voltage source, you will get it wrong as 6A. We have already found that it is 4A. 

That is why you got it as 20A for I1 after source transformation. You can calculate currents only in the external circuit, after source transformation.