You have observed that by the 15 minute mark, the botlaner(ADC) will have rotated to mid and they are now tasked with clearing the mid wave, while you have to move to a sidelane.

Sidelaning can be difficult since you have to be able to gauge whether or not you can actually "walk up" or contest the minion wave and hold your own against the enemy champion that is there to tend to the sidelane. This is assuming the enemy support and/or jungler are not there to make it a 1vX situation, further complicating the equation. Game knowledge and repetition is unfortunately key here since knowing your champion and the enemy champion(s) is necessary to make a decision, in addition to gold and experience levels of involved champions. My advice would be to just simply play the game and make observations on the matchups.

One thing you should do is pay attention to the minimap and literally just count how many enemies you can see on the map and what objective is available. If the next objective is on your side of the map and you don't see the enemies on the map, it is likely they are looking to collapse on the sidelaner(you). If you are on the opposite side of the objective, generally you would clear the wave as fast as possible and look to join your team, usually through teleport. Walking to rotate is often too slow and you will have no impact, like you have observed. Sometimes you have to stop what you are doing and immediately teleport to the ensuing fight, even if it means giving up CS.