I think you could happily do two days in Bursa and the rest of the time in Istanbul
You can get ferries to Bursa from Yeni Kapa with IDO or a couple of locations with Budo, which may be a quicker and pleasanter trip.
https://www.ido.com.tr/
https://budo.burulas.com.tr/
There are some nice spa hotels in Bursa so you can have a nice relaxing Turkish bath.
You can look on trip adviser for lots of things to do in both Bursa and Istanbul
What I would advise is taking the commuter ferries for a few lira up the Bosphorus rather than a crowded and expensive tourist trip.
Eat fish sandwiches (balick ekmek) near the Galata Bridge
Eat Iskender kabab with Grape Juice (Siyah) recommend
?SKENDER ® Yavuz ?skendero?lu the original Iskender, all the Turkish people I know rave about the Iskender restaurant near the old bus station in town, but I can't remember the name,
Kebapē? ?skender BURSA | ?skender 1867'den Beri is also acceptable. None of these restaurants serves alcohol.
Walk along the Bosphorus or along the Marama shore, you can see many of the old defensive walls of Constantipole.