Method 1

class BrowserHistory:
    #Two Stacks to save the history for back and forward usage
    #TC: O(1) for init and visit, O(steps) for back and forward
    #SC: O(n) fpr n input homepage

    def __init__(self, homepage: str):
        self.L = [homepage]
        self.idx = 0
        self.upperbound = 0

    def visit(self, url: str) -> None:
        if self.idx == len(self.L) - 1:
            self.L.append(url)
            self.idx += 1
        else:
            self.idx += 1
            self.L[self.idx] = url

        self.upperbound = self.idx

    def back(self, steps: int) -> str:
        self.idx = max(0, self.idx-steps)
        return self.L[self.idx]

    def forward(self, steps: int) -> str:
        self.idx = min(self.idx+steps, self.upperbound)
        return self.L[self.idx]

Method 2:

class BrowserHistory:
    #Use one stack to store browsing history.
    #Use a global idx to denote the current location
    #Define a variable upperbound
    #History beyond the upperbound should be not considered
    #TC: O(1) for all operations
    #SC: O(n) fpr n input homepage

    def __init__(self, homepage: str):
        self.L = [homepage]
        self.idx = 0
        self.upperbound = 0

    def visit(self, url: str) -> None:
        if self.idx == len(self.L) - 1:
            self.L.append(url)
            self.idx += 1
        else:
            self.idx += 1
            self.L[self.idx] = url

        self.upperbound = self.idx

    def back(self, steps: int) -> str:
        self.idx = max(0, self.idx-steps)
        return self.L[self.idx]

    def forward(self, steps: int) -> str:
        self.idx = min(self.idx+steps, self.upperbound)
        return self.L[self.idx]