The alternative to serial mode is parallel mode, which is Playwright's default mode.
Now, if you want to run your tests in parallel mode, you have to design them in a way that there are no dependencies between the tests.
For example, in the first test, I visit a URL and perform an action, and in the subsequent test, I need to perform a follow-up action based on the previous test's result.
Sounds like these tests are not independent. That means you won't be able to run them in parallel and have to rely on serial mode.
It's almost certainly possible to refactor your tests into a form where they are independent. Here are some general tips for achieving this:
Use Playwright's browser contexts to run the tests in isolation.
Have each test perform all steps necessary to set up the test's preconditions. For example, when you have multiple tests that require a user to be logged in and that user being on some products page, repeat the steps for logging the user in and navigating to the products page in each test.
Playwright let's you create fixtures that can be used to avoid repeating the same code in each test, while still performing the same steps/actions.
Avoid mutating the same shared application state in multiple tests. For example, when you have one test for adding items to a to-do list and another test for removing items from a to-do list, make sure that each test creates its own to-do list.