Dhaka is not for the faint-hearted. The Bangladeshi capital is an overcrowded, noisy, steamy assault on the senses – a riot of colour, rickshaw-driver-pick-me imprecation, the smell of dopiaza curry and balls of fried vegetable bhaji (fritters) and the roar of flame beneath giant galvanised kettles of cha.

And then there’s the traffic; our hotel is in upmarket Gulshan, about 10km north of Old Dhaka. When I ask the concierge for a taxi to the old town, he says the journey will take two hours. I assume he has misheard.

He hadn’t. Dhaka has no MTR equivalent, so most journeys in this city of 15 million people are taken by road, in a frenzy. Our taxi edges into the flurry of cars, rickshaws and wooden carts piled with bundles of cloth pushed by barefoot merchants. Children dodge bonnets and the occasional goat before banging on car windows to attract attention to whatever it is they are trying to sell: popcorn, jaggery date palm fudge or balloons.

There are no traffic lights in Dhaka, no “After you, please”, just policemen ineffec­tually waving sticks and blowing whistles at crossroads while the traffic pushes and bumps its way around them. And it really does bump – a gentle nudge is acceptable and it’s rare to see a vehicle with an undamaged bumper or a full set of intact lights.

Old Dhaka is worth the trouble of getting to, though. This ancient sector is built around a curve of the Buriganga River and wandering the Sadarghat docks is time spent seeing the city as it once was; the maelstrom of traffic feels far away. Here, oranges in crates are unloaded from boats, alert street urchins with matted hair dashing after any shiny orb that rolls across the stone flags of the dock, laughing and clutching their prize. Other children fish with a tangle of nets from the banks and boat-hands wander the ghats, looking for patrons to be ferried across the river – a far more efficient means of travel.

So it is that we come to be sitting on a small wooden boat while, around us, triple-decked ferries leer over tiny rafts and brightly painted traditional wooden barges vie for space with the dirty metal hulks of cargo boats.

Our little vessel bucks and rubs up against others until we meet the current in the centre of the river and are propelled smoothly down the oily black Buriganga. Dhaka’s store­houses and garment shops look different framed by water and further along we pass the white River Mosque and Armenian Church, calling worshippers with loud­speakers and bells.

We alight at Ahsan Manzil, a pink palace built in 1872 by Nawab Abdul Ghani on the site of an old French factory. This grand classical building had fallen into disrepair before it was restored in the 1980s. It is now a calm and colourful reminder of Old Dhaka, with photos and exhibits of the town as it was, family portraits, tableware and the skull of Feroz Jung, Ghani’s favourite elephant.

Bangladesh is not a destination for the traveller in a rush. The train taking us out of Dhaka clanks its way at a sedate 50km/h, heading for Srimangal, six hours to the north.

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The untidy fringes of the capital soon give way to bright green fields of rice and neat villages in which women wash yards of rainbow saris in streams while disconsolate skinny hump-backed cows look on.

Bangladesh is a major tea producer and the industry is based in Srimangal, its rolling hills covered with sturdy little bushes in measured rows.

In the wet summer months, the fields and factories are as riotous and crowded as a main road in Dhaka, the pickers all women with the small, delicate hands necessary for the job. Now, however, the estates are eerily quiet as we cycle the dusty lanes and stop to watch monkeys whooping as they swing between treetops or allow women carrying bundles of firewood on their heads to cross.

As we pass a tiny school, its children are playing a noisy game of musical chairs, kicking up dust and throwing each other to the floor while a teacher blows her whistle as effectively as the traffic policemen in the capital.

From a tiny shed we are served lunch: fresh fiery chilli onion bhajis, which must be held in paper bags because they are so hot, and seven-layer tea. This Willy Wonka-style infusion, gulped down across Bangladesh, has seven layers of colour and seven distinct tastes; it is delicious and sweet and helps me pedal faster as we continue the journey.

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Our host for three nights in Srimangal is Sultana, a friendly and rather autocratic woman whose small riverside guesthouse is recommended by Lonely Planet. She insists on meal times to the nearest minute, but what meals they are! Fresh aubergine and pumpkin from her garden dressed with coriander, eggs, chapati and porridge for breakfast, all washed down with pints of local tea.

Back in Dhaka at midnight we face our longest and slowest journey of all – a road trip to Khulna, in the south, that will take all night by bus. This journey is complicated by the huge rivers that make their way towards the Bay of Bengal; the bus has to be driven onto and then off a series of ferries.

Chasing tigers in the Sundarbans

Khulna, a scruffy fishing town and final resting place for many a rusty boat, is the gateway to the Sundarbans National Park, the world’s largest mangrove forest and a Unesco World Heritage site since 1997. The enormous network of interconnecting waterways stretches inland for 80km and is home to some uniquely Bangla wildlife: more than 300 species of bird, wild boars, otters, crocodiles, river dolphins, cobras and, most exciting of all, the largest wild population of the Bengal tiger. There are about 400 of these fearsome creatures in the Sundarbans, and they account for an average of 30 human deaths a year, the majority of those being fishermen, woodcutters or honey collectors.

Five of us, including guide, cook and skipper, board our small and tidy wooden boat, the MB Mawali, and set off down the wide river on a wild and remote three-day adventure.

Time falls away as we explore the Sundarbans, the calm waters and swaying trees soothing after the chaos of Dhaka. The night sky is peppered with bright stars and in the days, we untie the Mawali’s rowing boat to scull down the smaller channels, spotting birds and other wildlife. We see no tigers; the guide says he hasn’t spotted one in 10 years of leading people into the national park.

On our final day, we row silently to a beach at the southern tip of Bangladesh, where the bay opens into ocean that extends all the way to the ice of the Antarctic.

As we step onto dry land for the first time in three days, our hearts beat a little faster knowing tigers are out here somewhere. Boars, bigger than those in Hong Kong, run across the beach to the safety of the mangrove when they hear our approach and giant red-legged crabs scuttle sideways before disappearing into holes. Suddenly, there they are: fresh, clear pugmarks in the sand that disappear into the forest.

A tiger is out there somewhere ...