What you've done is incorporated a scrambling algorithm as part of your overall secret. But scrambling algorithms are not necessarily strong encryption. They're more like puzzles, and they can usually be solved by a dedicated attacker given enough skill, samples, and motivation. (For many examples of people who solve these kinds of problems just for fun, check out puzzles.stackexchange.com.) This leaves all your passwords vulnerable to the weakest sites you visit, not the strongest banking sites.
Let's first assume there are attackers who have the skills. Imagine an unemployed mathematician and puzzle solver, living in a country that's in turmoil, no job prospects, and nothing better to do. Next, let's give him some motivation to attack you - the attacker learns through a boast on facebook that you've got enough money in your retirement account to buy two Lamborghinis. And right now, with this question, you've just told him you use an algorithm to derive all your passwords. He looks up more information about you by following #SOreadytohelp on Twitter. He finds you work at Metacortex, and a bit more googling reveals that Metacortex hosts employee retirement accounts at retirementfundservices.com. So now the only thing standing in the thief's way of your millions is acquiring enough samples to break your algorithm.
I don't know your particular algorithm, but anything simple enough to keep in your head and apply quickly to derive a password is likely not going to survive our attacker if he has three samples; and he'd like a fourth to test his theory. More samples will obviously help him solve the puzzle quicker, with more confidence.
The number of samples needed to crack your algorithm is very important to your security, because your personal security now depends on the total security of all the sites where you use this algorithm, including the weakest of them. But unless you're also a mathematician and puzzle solver, you won't personally know what that number really is. Is it three? Five? If you personally need 20 samples to figure it out, that doesn't guarantee all attackers also need 20 samples to reverse engineer your algorithm - there are many clever people out there.
Assuming the attacker needs only four samples, that means he only has to break into the four weakest sites you visit. Or let's say you previously registered with Adobe, Sony, and Xbox, and those passwords all ended up in pastebin. Now he needs only one more sample. (Let's hope you didn't also sign up on Ashley Madison.) Perhaps the attacker googles you further and discovers you like breeding rabbits as a hobby, and he finds your accounts on rabbit-breeder.org and hobby-rabbit-breeding.com. You can't blame them for not being high security sites, because they have no reason to think they're protecting anything serious like bank accounts. He attacks the first, but they don't have any easy vulnerabilities. He attacks the second with some simple SQL injection, and recovers your password.
Once enough samples have been stolen, the attacker performs the analysis offline, providing no external evidence of success or failure. You simply wake up one morning, check your retirement balance, and discover your money's been transferred to an offshore bank.
You've already considered a purpose built password manager, which is still the best recommendation you're likely to receive.
Consider a hardware tool, like Yubikey or Mooltipass; something you can carry, and that syncs with your online password manager.
You can also successfully use your proposed system by segregating your risk into categories. Low risk sites, like rabbit-breeder.com, get one master password. Medium risk sites, anywhere you use a credit card, get a different master password. High risk sites, anywhere you have banking access, each get their own unique 15+ character password. So now you have to remember only a few passwords, still have good security, and don't need a cumbersome tool.