In the following extract, the first & last items are from American sources, and the middle one is from a British source.
The first two items support the fact that to give somebody a hand is used to mean "to help someone". The third item confirms that (especially in US usage), that may also be expressed as "to lend somebody a hand"; although less common, that usage is certainly not unknown — and would certainly be understood — in the UK.
Conversely, if one were asking for help, one might say "Would you please give me a hand with this?". The expression would certainly be "give me a hand", but the "me" might be partially 'swallowed' and not heard clearly.
If offering help, I might say "Do you want a hand with that?". It's easy to see that a non-native speaker may get confused between "give me a hand" and "Do you want a hand?".
(As an aside, the usage of give a big hand to someone (note the word "big" in the middle) to mean "give a round of applause" is also common in the UK.)
Finally, although the expression "I will help you" may be slightly non-idiomatic, I cannot think that by itself it should be so strange as to make people laugh; but it's possible that it might sound funny in the context of something said previously.