I am having this weird problem while playing league

My game acts like its lagging (high ping) but when I see my ping number its in the 30s. So when I see other player killing minions the minion health bar goes to 0 and disappear, while the minion death animation does not even happen after a few seconds. Even auto attack animation on minions and champions is not showing but I can see the health bar going to 0 and disapear. Movement animation is kind of the same thing but instead its jumping from one area to another. While this is all happening my ping is showing 35 36 33 38 and fps 40 41 45 48. Sometime my game disconnect during the game and I have to reconnect. Lastly, I check my internet speed while this is happening and it show 24 to 25 Mbps (Wired Connection). I am so confuse as to what is happening and if this doesnt get solve I cant play league anymore. Also, this doesnt happen the whole time it would be fine the first 5 min and then go bad and then get disconnected. I go back into the game and its perfectly fine. After a few min the problem happens again.
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