Count the islands

Scan every cell. When you find land you have not visited, that is a new island: count it, then visit all the land connected to it so you do not count it again.

def count_islands(grid):
    if not grid:
        return 0
    rows, cols = len(grid), len(grid[0])
    seen = set()
    islands = 0
    for r in range(rows):
        for c in range(cols):
            if grid[r][c] != "1" or (r, c) in seen:
                continue
            islands += 1
            stack = [(r, c)]
            seen.add((r, c))
            while stack:
                y, x = stack.pop()
                for ny, nx in ((y + 1, x), (y - 1, x), (y, x + 1), (y, x - 1)):
                    if 0 <= ny < rows and 0 <= nx < cols:
                        if grid[ny][nx] == "1" and (ny, nx) not in seen:
                            seen.add((ny, nx))
                            stack.append((ny, nx))
    return islands

The grid is really a graph: each land cell is a node joined to its land neighbours, and islands are connected components. The explicit stack does a depth-first search without recursion, so a huge island cannot hit Python's recursion limit. Each cell is handled a constant number of times: O(rows × cols).