I recently found more information about this, so I'll self-answer. While pesticide can have a big impact on the environment, the biggest problem with palm oil seems to be the clearing of rainforests.
This webpage says:
While organic farming prohibits the use of pesticides, chemical fertilizers and genetic engineering, organic labels do not establish standards against rainforest destruction and land grabbing. Furthermore, vast industrial monocultures – including those that fulfill organic production standards – can hardly be deemed environmentally sound.
and as an example they say that
Agro Palma – the largest palm oil producer in Brazil – states that it operates 39,000 hectares of oil palm plantations on former rainforest land in the state of Pará in the Amazon. Its certified organic palm oil is produced on the 4,107-hectare Hacienda CPA
This implies that forests have been cleared for organic palm plantations. Of course this is only 1 viewpoint, but I haven't been able to find any other sources that backup or refute the claims that are made.
So it seems LShaver is correct with his comment that organic-certified palm oil is no guarantee for a sustainable, low-environmental impact product.
The same article also says that the RSPO label is hardly a guarantee either;
The RSPO does not rule out the clearing of rainforest. Only primary and “high conservation value” (HCV) forests have been considered off-limits for palm oil plantations under the label since November 2005. An internationally recognized definition of HCV areas has not been established, however, and the transitions between primary and secondary rainforests are blurred in practice.
Rainforest Alliance label
The Rainforest Alliance says that
the clearing of natural forests is forbidden on Rainforest Alliance Certified oil palm plantations, which helps to protect the habitat of endangered wildlife such as orangutans. And farmers we work with who once relied on harmful pesticides and chemicals are now reducing or eliminating these harmful substances and replacing them with ecologically sound alternatives.
This treehugger blog confirms the above.
I haven't been able to find any reports that say that for palm oil Rainforest Alliance does not live up to their claims, so it looks like a Rainforest Alliance label is best of the 3 labels from an overall-sustainability point of view.
Boycotting palm oil all together is not a solution because companies would need to switch to alternative oils which will require more agricultural land
for an equivalent yield (see also https://www.huiledepalmedurable.org/le-boycott-est-il-la-solution/?lang=en)