I am from Bangladesh and I am visiting the USA in December. Is there any way I can get a transit visa for the UK and stay there for 4-5 days, before I go to the US?
A Visitor in Transit visa for the UK allows only stays of less than 48 hours. For a 4-5 days visit you need to apply for an ordinary Standard Visitor visa.
If you're traveling to or from USA with a valid US visa, you may qualify for landside Transit Without Visa in the UK, but only if there's at most one midnight coming between your arrival and departure, so it doesn't work for 4-5 days.