They started to get weaker when I started to support them.
"応援し始めた途端に弱くなり始めた" - They started to get weaker when I started to support them.
応援し始めた
- when I started to root for them
- When I became their fan
- When I started to support them
2000年代後半
- late 2000s
- the latter half of the 2000s
弱くなり始めた
- started to get weaker
- started to get worse
タイガースは、僕が応援し始めた2000年代後半から弱くなり始めた。
- I started to support the Tigers in the late 2000s, when they started to get worse.
- The Tigers started to get worse in the late 2000s when I began to support them.
They started playing badly as soon as I started cheering for them.
「応援し始めた途端に弱くなり始めた」は英語で「They started playing badly as soon as I started cheering for them.」と言います。「応援」と言うのは英語で「cheering」でも「supporting」でも言えます。
タイガースは、僕が応援し始めた2000年代後半から弱くなり始めた。
The Tigers started playing badly as soon as I started cheering for them in the second half of 2000.
アメリカ合衆国 
カナダ 
