If R1 has a NAT rule that is too specific, or a firewall which only allows 192.168.2.0/24 to go out to the Internet, then this is another problem.
While testing, disable any filter rules on R2 - take them out of the equation. Once you have the Internet working, add them back and if it causes problems, you will know exactly what the source of the problem is.
Anyway - on R1:
if it is a Mikrotik (never has been stated) then you should look in the NAT firewall table, on the srcnat chain.
Many people would have a rule like this:
action=masquerade src-address=192.168.2.0/24 action=masquerade
While this works for the first lan, it’s very specific.
The correct NAT rule (in my opinion) is this:
if ether1 = the wan interface of R1…
/ip firewall nat add chain=srcnat out-interface=ether1 action=masquerade
(and remove the old masquerade rule)
Also make sure the filter table will allow 192.168.3.0/24 → internet