Be careful what you wish for, you might get it!
You are asking for a non-floating figure ("Were I exactly want" = [H] option from float package.) That is OK, but if I understand correctly your question, when the image or table does not fit on the current page (and jump to the next page), actually do you want this object "exactly below where it is". In other words: you want a floating figure or a floating table!
Then, what do you want really is the classisc [h] option (lowercase, no packages required). When is not possible place the float [h]ere, LaTeX will try to put in the [t]op of the next page (i.e., it change the option to [ht] automatically).
Please experiment replacing [H] to [h] in this example reproducing your problem:
\documentclass[12pt]{article}
\usepackage{lipsum,color,float}
\usepackage{graphicx}
\begin{document}
\lipsum[1-2] % dummy black text *before* the image
\begin{figure}[H] % change to [h] to avoid the ugly result
\centering\includegraphics[height=12cm]{example-image-a}
\end{figure}
\color{blue}\lipsum[3-6] % dummy blue text *after* the image
\end{document}
Note that if LaTeX cannot do neither [h] in the actual page nor [t] in the next page, the float could be displaced more than one page (in the worst case, until the end of the document). Therefore, usually is a good idea allow flexible floats by default, for example with [!htbp] opions, and left to LaTeX find the best automatic solution. Only in the very final draft, if you are not pleased with the automatic results, change as few as possible options in as few as possible floats.
Required reading:
How to influence the position of float environments like figure and table in LaTeX?
Htells latex not to float the table so it never moves out of the position in the source text flow, so large spaces at page breaks are inevitable. The whole reason for allowing tables to float is to avoid those spaces. – David Carlisle Sep 30 '14 at 14:28\begin{figure}[!htp]. You can have a look at this question, to see why it is not recommended to use just[H]. – umarcor Sep 30 '14 at 15:27