I'd like to join the chorus but if everyone does, things get boring, so I choose not to:
Your knight got screwed.
He has a small island but no "anchor points" on either side of the river. The existing bridge likely already has an owner (There are other lords on either river bank, right?) who will be displeased to have the bridge taken away.
The king promised "the new lord will be able to collect plenty of taxes" probably without ensuring to keep it. After all, he said nothing about water rights. Well that's politics.
However his situation is not too bad, if he's clever (and a bit ruthless)
First he needs to find out the status of the bridge.
- Is it good or will it need repairs soon?
- Are there any claims on it and if yes, how strong is his claim in comparison?
Finding out the claims does not need to be easy. There's no central register. Any close city might have something in their records but a written contract in some private drawer might be lurking around, too.
In medieval times credits were usually paid back by nobles in terms of "royalties" (pun likely not coincidental but etymological). The noble got money now in exchange for lending his rights at something later like a source of tax income for a certain amount of time. This means the bridge as a tax source might be lend off to a rich family (the forerunner of banks) only to fall back to him after the time agreed in advance expires. The bank gambles on the income in the near future. They will calculate carefully to make sure they'll get their share.
This might already be the case. If so, the bank has a strong but time-limited claim. If not, any other claim (by other lords) can be circumvented:
The knight should address the king with concerns about the state and load of the bridge. He proposes to reduce the load on the already run-down (even if it's not) bridge to allow for repairs, if he gets in writing the right for taxing any new and old bridge located once the length from his island to the capital downstream and twice upstream (supposing the capital is located upstream, otherwise vice versa) regardless of previous arrangements.
If he's able to get that, he has royal decree that nullifies any claims and ensures he will always have the closest bridge to the capital. Plus he doesn't need to repair the old bridge, because he only assured to allow for not actually pay for repairs on his own.
Next he gets a credit lending out tax rights on the old bridge. The bank is bound to ask for the use of the money: "Constructions, I want to invest in my land to make it prosper by attracting more trade." The bank will likely agree, because more trade means more income from the bridge, which they will get.
Now the old bridge isn't of any concern for him. He already got a fixed share of the money beforehand. If there are any claims: Sort them out with the bank yourself!
He invests the money into building a new bridge. The bank will be pissed, because he'll be cutting their tax income with the new bridge: "But I said I'll invest in constructions!" With the two concurrent bridges, tax will be low attracting many merchants. The knight can increase the tax, as soon as the old bridge falls back to him.
In case there are strong claims over the old bridge, he can also sabotage it. He should only do so after the bank's claim expired. No need to anger them too much.
Now for water rights: He can have some bandits coming along the river. Attracting bandits shouldn't be too hard. If everything else fails, have some loyal men play them.
The bandits anger the king a little. And the knight reports his watch saw a boat escape along the river around that time. However he was unable to do anything and still is, unless he's allowed to deploy some chains across the river to stop and check travelers.
This should clarify water rights right away.
Now the Knight owns two sources of income:
- water rights
- new bridge (low tax at first)
He angered the bank but pleased the king (securing and extending an important transport route, defending against future assaults from the river) and merchants (low tax). Investing the income into building his castle step by step is the feasible way to go. It's unlikely he gets any credit after the stunt he pulled on the bank.
As an alternative, agree with the bank to finish the new bridge after their claim expired. This will keep them happy and taxes up. However it will attract less merchants.