when I write to the input, it stops doing focus

Problem

You have defined the NewUrlForm component inside the Home component. That is causing the focus issue.

Solution

1. Move the NewUrlForm as a separate component to the Home component level.

function NewUrlForm({ onSubmit, originalUrl, inputHandler }) {
  return (
    <>
      <form onSubmit={onSubmit}>
        <label
          className="block mb-5 text-lg font-medium text-gray-900 dark:text-gray-600"
          htmlFor="original-url"
        >
          URL
        </label>
        <input
          value={originalUrl}
          onChange={inputHandler}
          className="block w-full px-3 py-1.5 text-base font-normal text-gray-700 bg-white bg-clip-padding border border-solid border-gray-300 rounded transition ease-in-out m-0 focus:text-gray-700 focus:bg-white focus:border-blue-600 focus:outline-none
            "
          id="original-url"
          type="text"
          placeholder="https://example.com"
          required
        />

        <button
          type="submit"
          className="trasition duration-200 text-white bg-blue-700 hover:bg-blue-800 font-medium rounded-lg text-sm px-5 py-2.5 mr-2 mt-6 mb-2 w-full"
        >
          Create
        </button>
      </form>
    </>
  );
}

2. Pass the required props to NewUrlForm component.

<NewUrlForm
  onSubmit={onSubmit}
  originalUrl={originalUrl}
  inputHandler={inputHandler}
/>