Shopping online can indeed be more convenient. There are a few sites that i found here.
Bikroy
DarazBD
ajkerdeal
and more. These you can find by a simple google search.
My brief browsing shows me that Daraz is the most economical of them. atleast for what i was looking for.
Daraz is also a brand available and tested in my home country so i was inclined to buy from them further.
I found that small electronics are cheaper in Bangladesh sometimes, I got a desk lamp from Daraz which I Needed here and a few small lights. Other things like clothes are better to buy in person from various markets across Dhaka.
Daraz also has some flash sales often on their websites and sometimes the discounts are unbelievable, if they are on what you need then bingo!
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Delivery times are a longer than I was used to. In Dhaka they will take up-to four days to deliver. But that is the max. Mine was delivered on the fourth day. Also all items on your order may or may not be delivered together, Mine were delivered on the Third and Fourth day. But i checked and the product was good. It was new. It was as depicted in the advertisement. Daraz does not charge shipping for most of their listings. I was surprised to see they were delivering things as low as 30Tk for free.
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Best of luck.