It does not help our nature to just go shopping by bike. 

Supermarkets, markets and shops are giving away lightweight plastic bags for free. Those bags are mostly designed for just a single use. Because of their weight they can get blown away easily by wind and litter our environment. 

The problem is that those plastic bags are not biodegradable. They clog waterways, spoil the landscape, and end up in landfills where they may take 1,000 years or more to break down into ever smaller particles that continue to pollute the soil and water. Plastic bags are a serious danger to birds and marine mammals that often mistake them for food. Thousands die each year after swallowing or choking on discarded plastic bags.

Moreover for the production of those plastic bags millions of gallons of petroleum are required that could also be used for transportation or heating.

Many people try to act environmental friendly in their normal daily life but it seems like they are forgetting it during  grocery shopping. So the pollution someone eventually did not cause by going by bike to the supermarket is caused anyway already by the production of the plastic bags. 
The success of the plastic bag has meant a dramatic increase in the amount of bags found floating in the oceans where they choke, strangle, and starve wildlife and raft species around the world, according to David Barnes, a marine scientist with the British Antarctic Survey in Cambridge, England. Barnes said that plastic bags have gone "from being rare in the late 80s and early 90s to being almost everywhere from Spitsbergen 78° North [latitude] to Falklands 51° South [latitude], but I'll bet they'll be washing up in Antarctica within the decade."

So far many countries, cities and towns have introduced local bans on plastic bags. Their costumers are encouraged to bring bags with them when shopping. This ban is often reported as being a great success. Bans like this work best when the community has been involved in the decision making process and therefore feel a form of ownership of the project. So it is a fact that the banning of plastic bags works and now it is the part of the shops, governments to act but most important by the consumers to change their behavior.